| L(s) = 1 | − 4·2-s + 3·3-s + 16·4-s + 25·5-s − 12·6-s − 64·8-s − 234·9-s − 100·10-s + 405·11-s + 48·12-s + 391·13-s + 75·15-s + 256·16-s − 999·17-s + 936·18-s − 2.34e3·19-s + 400·20-s − 1.62e3·22-s + 2.43e3·23-s − 192·24-s + 625·25-s − 1.56e3·26-s − 1.43e3·27-s + 8.25e3·29-s − 300·30-s − 4.01e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.447·5-s − 0.136·6-s − 0.353·8-s − 0.962·9-s − 0.316·10-s + 1.00·11-s + 0.0962·12-s + 0.641·13-s + 0.0860·15-s + 1/4·16-s − 0.838·17-s + 0.680·18-s − 1.48·19-s + 0.223·20-s − 0.713·22-s + 0.957·23-s − 0.0680·24-s + 1/5·25-s − 0.453·26-s − 0.377·27-s + 1.82·29-s − 0.0608·30-s − 0.750·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p^{5} T^{2} \) |
| 11 | \( 1 - 405 T + p^{5} T^{2} \) |
| 13 | \( 1 - 391 T + p^{5} T^{2} \) |
| 17 | \( 1 + 999 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2342 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2430 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8259 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4016 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7042 T + p^{5} T^{2} \) |
| 41 | \( 1 + 3336 T + p^{5} T^{2} \) |
| 43 | \( 1 + 23518 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10317 T + p^{5} T^{2} \) |
| 53 | \( 1 - 3084 T + p^{5} T^{2} \) |
| 59 | \( 1 - 18816 T + p^{5} T^{2} \) |
| 61 | \( 1 + 21668 T + p^{5} T^{2} \) |
| 67 | \( 1 - 52124 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28560 T + p^{5} T^{2} \) |
| 73 | \( 1 - 70342 T + p^{5} T^{2} \) |
| 79 | \( 1 - 58823 T + p^{5} T^{2} \) |
| 83 | \( 1 + 756 T + p^{5} T^{2} \) |
| 89 | \( 1 + 135384 T + p^{5} T^{2} \) |
| 97 | \( 1 + 110435 T + p^{5} T^{2} \) |
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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
−9.561910017853326053759137647695, −8.612596001584738631398643227785, −8.455512631476923451027393046229, −6.72228596883890488555812744639, −6.40653777741689584979015214565, −5.06043265788305890726454122626, −3.66140012350253675739700761540, −2.47707829309665551466686833774, −1.38615340728587539547470615310, 0,
1.38615340728587539547470615310, 2.47707829309665551466686833774, 3.66140012350253675739700761540, 5.06043265788305890726454122626, 6.40653777741689584979015214565, 6.72228596883890488555812744639, 8.455512631476923451027393046229, 8.612596001584738631398643227785, 9.561910017853326053759137647695
