| L(s) = 1 | − 27·3-s + 160·5-s + 974·7-s + 729·9-s + 3.95e3·11-s + 574·13-s − 4.32e3·15-s − 8.47e3·17-s − 5.33e4·19-s − 2.62e4·21-s − 9.84e4·23-s − 5.25e4·25-s − 1.96e4·27-s + 5.10e4·29-s + 2.05e5·31-s − 1.06e5·33-s + 1.55e5·35-s + 2.55e5·37-s − 1.54e4·39-s − 6.65e5·41-s − 3.96e5·43-s + 1.16e5·45-s + 5.49e5·47-s + 1.25e5·49-s + 2.28e5·51-s − 7.20e5·53-s + 6.32e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.572·5-s + 1.07·7-s + 1/3·9-s + 0.896·11-s + 0.0724·13-s − 0.330·15-s − 0.418·17-s − 1.78·19-s − 0.619·21-s − 1.68·23-s − 0.672·25-s − 0.192·27-s + 0.388·29-s + 1.23·31-s − 0.517·33-s + 0.614·35-s + 0.829·37-s − 0.0418·39-s − 1.50·41-s − 0.761·43-s + 0.190·45-s + 0.771·47-s + 0.151·49-s + 0.241·51-s − 0.664·53-s + 0.512·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 - 32 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 974 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3956 T + p^{7} T^{2} \) |
| 13 | \( 1 - 574 T + p^{7} T^{2} \) |
| 17 | \( 1 + 8474 T + p^{7} T^{2} \) |
| 19 | \( 1 + 53312 T + p^{7} T^{2} \) |
| 23 | \( 1 + 98468 T + p^{7} T^{2} \) |
| 29 | \( 1 - 51060 T + p^{7} T^{2} \) |
| 31 | \( 1 - 205014 T + p^{7} T^{2} \) |
| 37 | \( 1 - 255674 T + p^{7} T^{2} \) |
| 41 | \( 1 + 665394 T + p^{7} T^{2} \) |
| 43 | \( 1 + 396840 T + p^{7} T^{2} \) |
| 47 | \( 1 - 549388 T + p^{7} T^{2} \) |
| 53 | \( 1 + 720060 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1043956 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2055734 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2092652 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2723868 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5190010 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3647110 T + p^{7} T^{2} \) |
| 83 | \( 1 - 2101956 T + p^{7} T^{2} \) |
| 89 | \( 1 - 522514 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1361990 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.881476566822259439326238890989, −8.695514772495158649578770779769, −7.950829976503085740103396913259, −6.55052171348461614746377125099, −6.02779841403731457024082510246, −4.75626735606133609431678418280, −4.04029661712800045175590321331, −2.18570072228670319492945878827, −1.43530153565680182224904712363, 0,
1.43530153565680182224904712363, 2.18570072228670319492945878827, 4.04029661712800045175590321331, 4.75626735606133609431678418280, 6.02779841403731457024082510246, 6.55052171348461614746377125099, 7.950829976503085740103396913259, 8.695514772495158649578770779769, 9.881476566822259439326238890989
