| L(s) = 1 | + 2.46·2-s − 3-s + 4.07·4-s − 5-s − 2.46·6-s − 4.49·7-s + 5.11·8-s + 9-s − 2.46·10-s + 1.05·11-s − 4.07·12-s + 2.71·13-s − 11.0·14-s + 15-s + 4.46·16-s + 3.79·17-s + 2.46·18-s − 4.64·19-s − 4.07·20-s + 4.49·21-s + 2.61·22-s − 5.11·24-s + 25-s + 6.69·26-s − 27-s − 18.3·28-s + 3.82·29-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.03·4-s − 0.447·5-s − 1.00·6-s − 1.69·7-s + 1.80·8-s + 0.333·9-s − 0.779·10-s + 0.319·11-s − 1.17·12-s + 0.753·13-s − 2.96·14-s + 0.258·15-s + 1.11·16-s + 0.919·17-s + 0.581·18-s − 1.06·19-s − 0.911·20-s + 0.980·21-s + 0.556·22-s − 1.04·24-s + 0.200·25-s + 1.31·26-s − 0.192·27-s − 3.46·28-s + 0.710·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.670132186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.670132186\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 7 | \( 1 + 4.49T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 + 0.381T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 0.415T + 41T^{2} \) |
| 43 | \( 1 - 8.21T + 43T^{2} \) |
| 47 | \( 1 + 4.78T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.35104668581140444589562529329, −6.67131890984862062549182769487, −6.43358721456056170632940755660, −5.68242995520087678520886736599, −5.14139977267800893670164195015, −4.03905428505539727933092364828, −3.77311159241844498404653413587, −3.09778342446527602851826918001, −2.16872961634526281799749229694, −0.72880082132995325290439169385,
0.72880082132995325290439169385, 2.16872961634526281799749229694, 3.09778342446527602851826918001, 3.77311159241844498404653413587, 4.03905428505539727933092364828, 5.14139977267800893670164195015, 5.68242995520087678520886736599, 6.43358721456056170632940755660, 6.67131890984862062549182769487, 7.35104668581140444589562529329
