| L(s) = 1 | + 32·2-s − 396·3-s + 7.35e3·5-s − 1.26e4·6-s − 3.27e4·8-s + 1.77e5·9-s + 2.35e5·10-s + 1.08e5·11-s + 1.27e6·13-s − 2.91e6·15-s − 1.04e6·16-s − 9.22e6·17-s + 5.66e6·18-s − 7.55e6·19-s + 3.48e6·22-s − 2.64e7·23-s + 1.29e7·24-s + 4.88e7·25-s + 4.06e7·26-s − 1.48e8·27-s − 3.39e8·29-s − 9.31e7·30-s − 5.13e7·31-s − 4.30e7·33-s − 2.95e8·34-s + 2.51e8·37-s − 2.41e8·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.940·3-s + 1.05·5-s − 0.665·6-s − 0.353·8-s + 9-s + 0.743·10-s + 0.203·11-s + 0.949·13-s − 0.989·15-s − 1/4·16-s − 1.57·17-s + 0.707·18-s − 0.700·19-s + 0.144·22-s − 0.858·23-s + 0.332·24-s + 25-s + 0.671·26-s − 1.98·27-s − 3.07·29-s − 0.699·30-s − 0.322·31-s − 0.191·33-s − 1.11·34-s + 0.596·37-s − 0.494·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.4988030264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4988030264\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_2$ | \( 1 - p^{5} T + p^{10} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 44 p^{2} T - 251 p^{4} T^{2} + 44 p^{13} T^{3} + p^{22} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 294 p^{2} T + 8311 p^{4} T^{2} - 294 p^{13} T^{3} + p^{22} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 108780 T - 273478582211 T^{2} - 108780 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 635842 T + p^{11} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 9225918 T + 50845666635091 T^{2} + 9225918 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 7555372 T - 59406612839835 T^{2} + 7555372 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26489400 T - 251121445553927 T^{2} + 26489400 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 169827594 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 51362704 T - 22770349534213215 T^{2} + 51362704 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 251605906 T - 114612089845379577 T^{2} - 251605906 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 928817814 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 1818895756 T + p^{11} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 523343136 T - 2198271177085697807 T^{2} - 523343136 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4199520078 T + 8366932956152934487 T^{2} + 4199520078 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 154917444 p T + 15336418636838197 p^{2} T^{2} - 154917444 p^{12} T^{3} + p^{22} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6639312802 T + 566556871365252543 T^{2} + 6639312802 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2878139188 T - \)\(11\!\cdots\!39\)\( T^{2} - 2878139188 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4345596360 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 23450332826 T + \)\(23\!\cdots\!99\)\( T^{2} - 23450332826 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 28761853648 T + 79250414741449979025 T^{2} - 28761853648 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 5577757548 T + p^{11} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 78002173386 T + \)\(33\!\cdots\!07\)\( T^{2} - 78002173386 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 26685859630 T + p^{11} T^{2} )^{2} \) |
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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
−12.77499067863489095778932900187, −11.27070439859897224250080298079, −11.12654797530554112049290677344, −10.62149105969894929014112283751, −9.755635028373877927984090844220, −9.430592567842418704787787401084, −8.951157409759220364530487580500, −8.129558058104097529465156902447, −7.38340651081392684834870044439, −6.65870112328431451788996550235, −6.11679224482734745609628844624, −5.94493633483124887003304093402, −5.18706282801328196236293315642, −4.66098445756697882203369114424, −3.85132848559974115127018772846, −3.61141853388469553346131318240, −2.07922621717804686222116671841, −2.07393754172117069496895141245, −1.21865433188877782805500490675, −0.15500098466309583674697467108,
0.15500098466309583674697467108, 1.21865433188877782805500490675, 2.07393754172117069496895141245, 2.07922621717804686222116671841, 3.61141853388469553346131318240, 3.85132848559974115127018772846, 4.66098445756697882203369114424, 5.18706282801328196236293315642, 5.94493633483124887003304093402, 6.11679224482734745609628844624, 6.65870112328431451788996550235, 7.38340651081392684834870044439, 8.129558058104097529465156902447, 8.951157409759220364530487580500, 9.430592567842418704787787401084, 9.755635028373877927984090844220, 10.62149105969894929014112283751, 11.12654797530554112049290677344, 11.27070439859897224250080298079, 12.77499067863489095778932900187
