| L(s) = 1 | + 2.38·2-s − 3.22·3-s + 3.68·4-s − 5-s − 7.68·6-s − 7-s + 4.00·8-s + 7.38·9-s − 2.38·10-s + 1.35·11-s − 11.8·12-s − 0.636·13-s − 2.38·14-s + 3.22·15-s + 2.18·16-s − 5.67·17-s + 17.6·18-s + 1.99·19-s − 3.68·20-s + 3.22·21-s + 3.23·22-s + 3.12·23-s − 12.9·24-s + 25-s − 1.51·26-s − 14.1·27-s − 3.68·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 1.86·3-s + 1.84·4-s − 0.447·5-s − 3.13·6-s − 0.377·7-s + 1.41·8-s + 2.46·9-s − 0.753·10-s + 0.409·11-s − 3.42·12-s − 0.176·13-s − 0.636·14-s + 0.832·15-s + 0.545·16-s − 1.37·17-s + 4.14·18-s + 0.458·19-s − 0.822·20-s + 0.703·21-s + 0.690·22-s + 0.651·23-s − 2.63·24-s + 0.200·25-s − 0.297·26-s − 2.72·27-s − 0.695·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 3 | \( 1 + 3.22T + 3T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 + 0.636T + 13T^{2} \) |
| 17 | \( 1 + 5.67T + 17T^{2} \) |
| 19 | \( 1 - 1.99T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 - 2.15T + 41T^{2} \) |
| 43 | \( 1 + 5.25T + 43T^{2} \) |
| 47 | \( 1 + 7.96T + 47T^{2} \) |
| 53 | \( 1 - 3.07T + 53T^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 - 8.00T + 67T^{2} \) |
| 71 | \( 1 + 3.75T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 1.21T + 83T^{2} \) |
| 89 | \( 1 - 9.93T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−6.96988939369729656319471583247, −6.40834916685069271156447043307, −6.16192641782395013794175526775, −5.19740244539378500477231441500, −4.76996806759161353804968995344, −4.24937639021639035423352682499, −3.48526283709132854762062405364, −2.46018869785511935195508511586, −1.23975613147116154660866768508, 0,
1.23975613147116154660866768508, 2.46018869785511935195508511586, 3.48526283709132854762062405364, 4.24937639021639035423352682499, 4.76996806759161353804968995344, 5.19740244539378500477231441500, 6.16192641782395013794175526775, 6.40834916685069271156447043307, 6.96988939369729656319471583247
