| L(s) = 1 | − 2.65e15·2-s − 2.23e24·3-s + 4.53e30·4-s + 2.85e35·5-s + 5.95e39·6-s + 3.30e41·7-s − 5.31e45·8-s + 3.46e48·9-s − 7.60e50·10-s − 4.96e52·11-s − 1.01e55·12-s + 5.64e55·13-s − 8.78e56·14-s − 6.40e59·15-s + 2.62e60·16-s − 1.94e61·17-s − 9.22e63·18-s + 3.52e64·19-s + 1.29e66·20-s − 7.39e65·21-s + 1.32e68·22-s − 1.63e68·23-s + 1.18e70·24-s + 4.23e70·25-s − 1.50e71·26-s − 4.30e72·27-s + 1.49e72·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s − 1.80·3-s + 1.78·4-s + 1.43·5-s + 3.00·6-s + 0.0694·7-s − 1.31·8-s + 2.24·9-s − 2.40·10-s − 1.27·11-s − 3.22·12-s + 0.314·13-s − 0.115·14-s − 2.59·15-s + 0.409·16-s − 0.141·17-s − 3.74·18-s + 0.932·19-s + 2.57·20-s − 0.125·21-s + 2.12·22-s − 0.279·23-s + 2.37·24-s + 1.07·25-s − 0.524·26-s − 2.24·27-s + 0.124·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(102-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+50.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(51)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{103}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.65e15T + 2.53e30T^{2} \) |
| 3 | \( 1 + 2.23e24T + 1.54e48T^{2} \) |
| 5 | \( 1 - 2.85e35T + 3.94e70T^{2} \) |
| 7 | \( 1 - 3.30e41T + 2.26e85T^{2} \) |
| 11 | \( 1 + 4.96e52T + 1.51e105T^{2} \) |
| 13 | \( 1 - 5.64e55T + 3.22e112T^{2} \) |
| 17 | \( 1 + 1.94e61T + 1.88e124T^{2} \) |
| 19 | \( 1 - 3.52e64T + 1.42e129T^{2} \) |
| 23 | \( 1 + 1.63e68T + 3.42e137T^{2} \) |
| 29 | \( 1 + 5.40e73T + 5.03e147T^{2} \) |
| 31 | \( 1 + 3.00e75T + 4.24e150T^{2} \) |
| 37 | \( 1 - 2.08e79T + 2.44e158T^{2} \) |
| 41 | \( 1 - 6.19e80T + 7.78e162T^{2} \) |
| 43 | \( 1 + 1.29e82T + 9.55e164T^{2} \) |
| 47 | \( 1 - 3.59e84T + 7.61e168T^{2} \) |
| 53 | \( 1 + 9.51e86T + 1.41e174T^{2} \) |
| 59 | \( 1 - 2.90e89T + 7.17e178T^{2} \) |
| 61 | \( 1 - 9.18e89T + 2.08e180T^{2} \) |
| 67 | \( 1 + 1.99e92T + 2.71e184T^{2} \) |
| 71 | \( 1 + 1.96e93T + 9.48e186T^{2} \) |
| 73 | \( 1 - 2.27e94T + 1.56e188T^{2} \) |
| 79 | \( 1 - 2.24e95T + 4.57e191T^{2} \) |
| 83 | \( 1 - 1.11e96T + 6.71e193T^{2} \) |
| 89 | \( 1 + 1.83e98T + 7.73e196T^{2} \) |
| 97 | \( 1 - 2.50e100T + 4.61e200T^{2} \) |
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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
−13.05726882317307182855147887329, −11.26648236080581207087359860845, −10.38300120215739225791751469515, −9.492443287025581014575019459167, −7.47377541199048210675539711983, −6.14685924467856461217161361031, −5.25247243719148982095164323036, −2.10150571005205108207446336184, −1.06379456386511640564012204306, 0,
1.06379456386511640564012204306, 2.10150571005205108207446336184, 5.25247243719148982095164323036, 6.14685924467856461217161361031, 7.47377541199048210675539711983, 9.492443287025581014575019459167, 10.38300120215739225791751469515, 11.26648236080581207087359860845, 13.05726882317307182855147887329
