| L(s) = 1 | + 0.242·2-s − 1.19·3-s − 1.94·4-s − 0.289·6-s − 4.78·7-s − 0.957·8-s − 1.57·9-s + 1.30·11-s + 2.31·12-s − 1.16·14-s + 3.64·16-s + 5.83·17-s − 0.382·18-s − 4.27·19-s + 5.70·21-s + 0.316·22-s + 4.13·23-s + 1.14·24-s + 5.46·27-s + 9.27·28-s − 3.08·29-s + 5.43·31-s + 2.80·32-s − 1.55·33-s + 1.41·34-s + 3.05·36-s + 4.84·37-s + ⋯ |
| L(s) = 1 | + 0.171·2-s − 0.689·3-s − 0.970·4-s − 0.118·6-s − 1.80·7-s − 0.338·8-s − 0.525·9-s + 0.392·11-s + 0.668·12-s − 0.310·14-s + 0.912·16-s + 1.41·17-s − 0.0901·18-s − 0.980·19-s + 1.24·21-s + 0.0673·22-s + 0.862·23-s + 0.233·24-s + 1.05·27-s + 1.75·28-s − 0.573·29-s + 0.975·31-s + 0.495·32-s − 0.270·33-s + 0.243·34-s + 0.509·36-s + 0.795·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.242T + 2T^{2} \) |
| 3 | \( 1 + 1.19T + 3T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 + 3.08T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 - 4.84T + 37T^{2} \) |
| 41 | \( 1 + 8.00T + 41T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 - 6.31T + 47T^{2} \) |
| 53 | \( 1 + 0.506T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 + 4.07T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 + 7.18T + 79T^{2} \) |
| 83 | \( 1 - 9.99T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 2.82T + 97T^{2} \) |
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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
−8.128490349015451971082741532730, −7.14341640292468582543202822294, −6.25798535648818237687410942420, −5.93741637162468663784769698843, −5.15311705329083808914860018611, −4.23438466274922715418736224465, −3.40117849762344025841821205488, −2.82675568354778218327278732786, −0.947868956406344699208139504372, 0,
0.947868956406344699208139504372, 2.82675568354778218327278732786, 3.40117849762344025841821205488, 4.23438466274922715418736224465, 5.15311705329083808914860018611, 5.93741637162468663784769698843, 6.25798535648818237687410942420, 7.14341640292468582543202822294, 8.128490349015451971082741532730
