| L(s) = 1 | + 0.618·2-s − 2.61·3-s − 1.61·4-s − 1.61·6-s − 3·7-s − 2.23·8-s + 3.85·9-s + 0.618·11-s + 4.23·12-s + 13-s − 1.85·14-s + 1.85·16-s − 5.23·17-s + 2.38·18-s + 7.85·21-s + 0.381·22-s − 7.61·23-s + 5.85·24-s + 0.618·26-s − 2.23·27-s + 4.85·28-s − 1.38·29-s − 2.14·31-s + 5.61·32-s − 1.61·33-s − 3.23·34-s − 6.23·36-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 1.51·3-s − 0.809·4-s − 0.660·6-s − 1.13·7-s − 0.790·8-s + 1.28·9-s + 0.186·11-s + 1.22·12-s + 0.277·13-s − 0.495·14-s + 0.463·16-s − 1.26·17-s + 0.561·18-s + 1.71·21-s + 0.0814·22-s − 1.58·23-s + 1.19·24-s + 0.121·26-s − 0.430·27-s + 0.917·28-s − 0.256·29-s − 0.385·31-s + 0.993·32-s − 0.281·33-s − 0.554·34-s − 1.03·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 9.32T + 53T^{2} \) |
| 59 | \( 1 + 15.3T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 0.472T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 7.14T + 97T^{2} \) |
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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
−6.63499114485165053461344075340, −6.19113353714644464931651932211, −5.81207217659286280836400848484, −4.98139648877440494518488031172, −4.33240300002455137954587219552, −3.77138959132062611814378260721, −2.83510260492345328558137922831, −1.49185935022862112481008762478, 0, 0,
1.49185935022862112481008762478, 2.83510260492345328558137922831, 3.77138959132062611814378260721, 4.33240300002455137954587219552, 4.98139648877440494518488031172, 5.81207217659286280836400848484, 6.19113353714644464931651932211, 6.63499114485165053461344075340
