| L(s) = 1 | + 4·2-s − 19.4·3-s + 16·4-s − 36.7·5-s − 77.9·6-s + 204.·7-s + 64·8-s + 136.·9-s − 146.·10-s + 704.·11-s − 311.·12-s − 718.·13-s + 816.·14-s + 715.·15-s + 256·16-s − 1.84e3·17-s + 546.·18-s − 14.9·19-s − 587.·20-s − 3.97e3·21-s + 2.81e3·22-s − 1.24e3·24-s − 1.77e3·25-s − 2.87e3·26-s + 2.07e3·27-s + 3.26e3·28-s + 2.96e3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.24·3-s + 0.5·4-s − 0.656·5-s − 0.883·6-s + 1.57·7-s + 0.353·8-s + 0.562·9-s − 0.464·10-s + 1.75·11-s − 0.624·12-s − 1.17·13-s + 1.11·14-s + 0.820·15-s + 0.250·16-s − 1.54·17-s + 0.397·18-s − 0.00953·19-s − 0.328·20-s − 1.96·21-s + 1.24·22-s − 0.441·24-s − 0.568·25-s − 0.833·26-s + 0.546·27-s + 0.787·28-s + 0.655·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 19.4T + 243T^{2} \) |
| 5 | \( 1 + 36.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 204.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 704.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 718.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.84e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 14.9T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.51e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.32e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.86e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.50e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.05e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.48e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.81e4T + 8.58e9T^{2} \) |
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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
−8.631249478825411970777402616959, −7.72917229502641559929227636429, −6.80489954829061762226670457585, −6.22424532417916487257879492922, −4.99655421134761404768986208190, −4.65954468223570258341400498005, −3.85229506564377259769667481524, −2.20818804611360809661663714286, −1.22000232525028198634011706567, 0,
1.22000232525028198634011706567, 2.20818804611360809661663714286, 3.85229506564377259769667481524, 4.65954468223570258341400498005, 4.99655421134761404768986208190, 6.22424532417916487257879492922, 6.80489954829061762226670457585, 7.72917229502641559929227636429, 8.631249478825411970777402616959
