| L(s) = 1 | + 1.02e3·2-s + 3.37e4·3-s + 7.86e5·4-s + 3.90e6·5-s + 3.45e7·6-s + 8.30e7·7-s + 5.36e8·8-s − 3.01e8·9-s + 4.00e9·10-s + 3.54e9·11-s + 2.65e10·12-s + 3.02e10·13-s + 8.50e10·14-s + 1.31e11·15-s + 3.43e11·16-s + 3.29e11·17-s − 3.08e11·18-s + 3.83e11·19-s + 3.07e12·20-s + 2.80e12·21-s + 3.63e12·22-s + 2.08e12·23-s + 1.81e13·24-s + 1.14e13·25-s + 3.09e13·26-s − 1.94e13·27-s + 6.53e13·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.989·3-s + 3/2·4-s + 0.894·5-s + 1.39·6-s + 0.777·7-s + 1.41·8-s − 0.259·9-s + 1.26·10-s + 0.453·11-s + 1.48·12-s + 0.791·13-s + 1.10·14-s + 0.884·15-s + 5/4·16-s + 0.674·17-s − 0.366·18-s + 0.272·19-s + 1.34·20-s + 0.769·21-s + 0.641·22-s + 0.241·23-s + 1.39·24-s + 3/5·25-s + 1.11·26-s − 0.491·27-s + 1.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(16.05222287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.05222287\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{9} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{9} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 33724 T + 53278114 p^{3} T^{2} - 33724 p^{19} T^{3} + p^{38} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 83061292 T + 496307663656398 p^{2} T^{2} - 83061292 p^{19} T^{3} + p^{38} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3549480144 T + 6780691163172962306 p T^{2} - 3549480144 p^{19} T^{3} + p^{38} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2326940428 p T + 1384454580615323262 p^{2} T^{2} - 2326940428 p^{20} T^{3} + p^{38} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 19391516436 p T + \)\(25\!\cdots\!78\)\( p^{2} T^{2} - 19391516436 p^{20} T^{3} + p^{38} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 383076809800 T + \)\(37\!\cdots\!58\)\( T^{2} - 383076809800 p^{19} T^{3} + p^{38} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2082856096884 T + \)\(10\!\cdots\!38\)\( T^{2} - 2082856096884 p^{19} T^{3} + p^{38} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 166587684602220 T + \)\(15\!\cdots\!38\)\( T^{2} - 166587684602220 p^{19} T^{3} + p^{38} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 418151119038664 T + \)\(87\!\cdots\!66\)\( T^{2} - 418151119038664 p^{19} T^{3} + p^{38} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1292219637694628 T + \)\(15\!\cdots\!42\)\( T^{2} + 1292219637694628 p^{19} T^{3} + p^{38} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3127192714527996 T + \)\(48\!\cdots\!26\)\( T^{2} + 3127192714527996 p^{19} T^{3} + p^{38} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1795992338357636 T + \)\(56\!\cdots\!38\)\( T^{2} + 1795992338357636 p^{19} T^{3} + p^{38} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6761653310165988 T + \)\(11\!\cdots\!02\)\( T^{2} + 6761653310165988 p^{19} T^{3} + p^{38} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9787568571066516 T + \)\(11\!\cdots\!98\)\( T^{2} + 9787568571066516 p^{19} T^{3} + p^{38} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 25484316660294360 T + \)\(89\!\cdots\!78\)\( T^{2} + 25484316660294360 p^{19} T^{3} + p^{38} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 77840503559636276 T + \)\(11\!\cdots\!26\)\( T^{2} + 77840503559636276 p^{19} T^{3} + p^{38} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 439654835073204532 T + \)\(14\!\cdots\!62\)\( T^{2} - 439654835073204532 p^{19} T^{3} + p^{38} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 29427662567526696 T + \)\(15\!\cdots\!66\)\( T^{2} + 29427662567526696 p^{19} T^{3} + p^{38} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1463308911724217756 T + \)\(10\!\cdots\!58\)\( T^{2} + 1463308911724217756 p^{19} T^{3} + p^{38} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1235969155825112720 T + \)\(13\!\cdots\!38\)\( T^{2} + 1235969155825112720 p^{19} T^{3} + p^{38} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 877205635969635396 T + \)\(54\!\cdots\!98\)\( T^{2} + 877205635969635396 p^{19} T^{3} + p^{38} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4172929077000032820 T + \)\(20\!\cdots\!18\)\( T^{2} - 4172929077000032820 p^{19} T^{3} + p^{38} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11254578228347861332 T + \)\(10\!\cdots\!22\)\( T^{2} - 11254578228347861332 p^{19} T^{3} + p^{38} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.05474374817133099503041584347, −15.53385468812915918999060100400, −14.47592204804791169752918476014, −14.33612100989044385513149005056, −13.70588276651564951966614978321, −13.28037453479219692582828796213, −12.03541337679875878004693680349, −11.70206648653484502223273437583, −10.53728891244057380236854088944, −9.860578206753873640962656382450, −8.444043481239405089174898302199, −8.277942908096038810030081546249, −6.75852607962359498526493161754, −6.19803253713975762842003915950, −5.12966342028173067685907704767, −4.53548926144809339788761660784, −3.13243784244599671360735054559, −2.98943631278692431912506676383, −1.75584299494003521194763748316, −1.18446682544636789798032429924,
1.18446682544636789798032429924, 1.75584299494003521194763748316, 2.98943631278692431912506676383, 3.13243784244599671360735054559, 4.53548926144809339788761660784, 5.12966342028173067685907704767, 6.19803253713975762842003915950, 6.75852607962359498526493161754, 8.277942908096038810030081546249, 8.444043481239405089174898302199, 9.860578206753873640962656382450, 10.53728891244057380236854088944, 11.70206648653484502223273437583, 12.03541337679875878004693680349, 13.28037453479219692582828796213, 13.70588276651564951966614978321, 14.33612100989044385513149005056, 14.47592204804791169752918476014, 15.53385468812915918999060100400, 16.05474374817133099503041584347
