| L(s) = 1 | − 2.56·3-s + 1.56·5-s + 7-s + 3.56·9-s − 4.56·11-s + 13-s − 4·15-s − 4·17-s + 6.68·19-s − 2.56·21-s − 5·23-s − 2.56·25-s − 1.43·27-s + 5.56·29-s − 6.12·31-s + 11.6·33-s + 1.56·35-s + 5.43·37-s − 2.56·39-s − 8.56·41-s + 11.8·43-s + 5.56·45-s − 3·47-s + 49-s + 10.2·51-s − 8.68·53-s − 7.12·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.698·5-s + 0.377·7-s + 1.18·9-s − 1.37·11-s + 0.277·13-s − 1.03·15-s − 0.970·17-s + 1.53·19-s − 0.558·21-s − 1.04·23-s − 0.512·25-s − 0.276·27-s + 1.03·29-s − 1.09·31-s + 2.03·33-s + 0.263·35-s + 0.894·37-s − 0.410·39-s − 1.33·41-s + 1.80·43-s + 0.829·45-s − 0.437·47-s + 0.142·49-s + 1.43·51-s − 1.19·53-s − 0.960·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 6.68T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 - 5.43T + 37T^{2} \) |
| 41 | \( 1 + 8.56T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 8.68T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 - 2.24T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 5.56T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−9.382802267916208423096094718655, −8.165906853775664844037350828666, −7.41929717545831737658859445728, −6.40010284461581982685235504280, −5.71433433428968956342271940488, −5.18956480407892960224862836609, −4.33748366969948651895993684939, −2.77950877325778868387542776997, −1.52039988183722188360753841505, 0,
1.52039988183722188360753841505, 2.77950877325778868387542776997, 4.33748366969948651895993684939, 5.18956480407892960224862836609, 5.71433433428968956342271940488, 6.40010284461581982685235504280, 7.41929717545831737658859445728, 8.165906853775664844037350828666, 9.382802267916208423096094718655
