| L(s) = 1 | + 1.25e3·5-s − 380·7-s − 1.02e5·11-s + 1.79e5·13-s − 3.16e5·17-s + 1.37e5·19-s + 6.65e5·23-s + 1.17e6·25-s + 6.89e6·29-s + 2.91e5·31-s − 4.75e5·35-s + 1.12e7·37-s − 2.97e7·41-s − 1.17e7·43-s − 6.24e7·47-s + 1.56e7·49-s − 9.41e6·53-s − 1.28e8·55-s + 9.29e7·59-s + 1.95e8·61-s + 2.23e8·65-s − 2.19e8·67-s − 3.11e8·71-s − 9.92e7·73-s + 3.90e7·77-s + 5.42e8·79-s + 1.25e9·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.0598·7-s − 2.11·11-s + 1.73·13-s − 0.917·17-s + 0.241·19-s + 0.495·23-s + 3/5·25-s + 1.80·29-s + 0.0567·31-s − 0.0535·35-s + 0.987·37-s − 1.64·41-s − 0.522·43-s − 1.86·47-s + 0.388·49-s − 0.163·53-s − 1.89·55-s + 0.998·59-s + 1.80·61-s + 1.55·65-s − 1.33·67-s − 1.45·71-s − 0.408·73-s + 0.126·77-s + 1.56·79-s + 2.90·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.785816613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.785816613\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 380 T - 2220450 p T^{2} + 380 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 102720 T + 7335543382 T^{2} + 102720 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1060 p^{2} T + 22798610142 T^{2} - 1060 p^{11} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 316020 T + 259693705798 T^{2} + 316020 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 137272 T + 111610161654 T^{2} - 137272 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 665460 T + 2886450615250 T^{2} - 665460 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6893748 T + 40195999658014 T^{2} - 6893748 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 291832 T + 38964935800398 T^{2} - 291832 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11261380 T + 218879982937230 T^{2} - 11261380 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 29773452 T + 771012402449398 T^{2} + 29773452 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 11708180 T + 838769843899386 T^{2} + 11708180 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 62493300 T + 3177958884734338 T^{2} + 62493300 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9417780 T + 5708185761526990 T^{2} + 9417780 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 92930856 T + 16656477955483462 T^{2} - 92930856 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 195673924 T + 22434263296171326 T^{2} - 195673924 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 219767420 T + 65652945987990090 T^{2} + 219767420 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 311207016 T + 76405636625293726 T^{2} + 311207016 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 99224060 T + 35402447061205782 T^{2} + 99224060 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 542261776 T + 313115996157615582 T^{2} - 542261776 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1256915700 T + 768086791626261130 T^{2} - 1256915700 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 462291852 T + 159603168035249494 T^{2} - 462291852 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1671716740 T + 2048690578856969670 T^{2} - 1671716740 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−11.04115131705730604604559046270, −10.71266000093042202950696753968, −10.16305367744356204892571373291, −10.02967871339269139522557391264, −9.128761487818169109817170471913, −8.734107315910264722179841313724, −8.130125615669988024223430332740, −7.969701946274284367246636534655, −6.91114814128166393906382219652, −6.60563748729831679357172720244, −5.99910974133467636015577508616, −5.50791507519377967773874353063, −4.83728718943836464606876265653, −4.57296354085014674373685383938, −3.28962010403200739373232580122, −3.18993156482060904667262149070, −2.25034043059916548410363031207, −1.90154876739565485109956218788, −0.945763263855134365869180153532, −0.50319341229780840412361566084,
0.50319341229780840412361566084, 0.945763263855134365869180153532, 1.90154876739565485109956218788, 2.25034043059916548410363031207, 3.18993156482060904667262149070, 3.28962010403200739373232580122, 4.57296354085014674373685383938, 4.83728718943836464606876265653, 5.50791507519377967773874353063, 5.99910974133467636015577508616, 6.60563748729831679357172720244, 6.91114814128166393906382219652, 7.969701946274284367246636534655, 8.130125615669988024223430332740, 8.734107315910264722179841313724, 9.128761487818169109817170471913, 10.02967871339269139522557391264, 10.16305367744356204892571373291, 10.71266000093042202950696753968, 11.04115131705730604604559046270
