| L(s) = 1 | − 3.48·2-s + 0.850·3-s + 4.14·4-s − 2.96·6-s + 13.4·8-s − 26.2·9-s − 6.90·11-s + 3.52·12-s − 22.1·13-s − 79.9·16-s + 88.3·17-s + 91.5·18-s − 36.9·19-s + 24.0·22-s + 95.5·23-s + 11.4·24-s + 77.1·26-s − 45.2·27-s + 269.·29-s − 197.·31-s + 171.·32-s − 5.87·33-s − 307.·34-s − 109.·36-s − 2.14·37-s + 128.·38-s − 18.8·39-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.163·3-s + 0.518·4-s − 0.201·6-s + 0.593·8-s − 0.973·9-s − 0.189·11-s + 0.0848·12-s − 0.472·13-s − 1.24·16-s + 1.25·17-s + 1.19·18-s − 0.446·19-s + 0.233·22-s + 0.866·23-s + 0.0970·24-s + 0.582·26-s − 0.322·27-s + 1.72·29-s − 1.14·31-s + 0.946·32-s − 0.0309·33-s − 1.55·34-s − 0.504·36-s − 0.00953·37-s + 0.549·38-s − 0.0772·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.48T + 8T^{2} \) |
| 3 | \( 1 - 0.850T + 27T^{2} \) |
| 11 | \( 1 + 6.90T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 95.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 2.14T + 5.06e4T^{2} \) |
| 41 | \( 1 + 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 17.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 528.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 641.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 142.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 478.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 105.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 986.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 711.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 636.T + 9.12e5T^{2} \) |
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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
−8.755048419024961319079000053421, −8.403778952311968924085388968649, −7.54432001149148617985425612627, −6.77411094534285200555074742729, −5.58684692238629444048452466254, −4.75293670393782785832007076085, −3.38894782765726875959368024079, −2.36451061647254012302596130086, −1.09618410474767050622916957760, 0,
1.09618410474767050622916957760, 2.36451061647254012302596130086, 3.38894782765726875959368024079, 4.75293670393782785832007076085, 5.58684692238629444048452466254, 6.77411094534285200555074742729, 7.54432001149148617985425612627, 8.403778952311968924085388968649, 8.755048419024961319079000053421
