| L(s) = 1 | − 306.·2-s + 1.66e3·3-s + 6.14e4·4-s + 6.40e4·5-s − 5.11e5·6-s + 1.64e6·7-s − 8.79e6·8-s − 1.15e7·9-s − 1.96e7·10-s − 6.23e7·11-s + 1.02e8·12-s − 1.11e8·13-s − 5.06e8·14-s + 1.06e8·15-s + 6.86e8·16-s + 3.06e9·17-s + 3.55e9·18-s + 2.24e8·19-s + 3.93e9·20-s + 2.74e9·21-s + 1.91e10·22-s + 3.40e9·23-s − 1.46e10·24-s − 2.64e10·25-s + 3.41e10·26-s − 4.32e10·27-s + 1.01e11·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.440·3-s + 1.87·4-s + 0.366·5-s − 0.746·6-s + 0.756·7-s − 1.48·8-s − 0.806·9-s − 0.621·10-s − 0.963·11-s + 0.824·12-s − 0.492·13-s − 1.28·14-s + 0.161·15-s + 0.639·16-s + 1.81·17-s + 1.36·18-s + 0.0576·19-s + 0.687·20-s + 0.333·21-s + 1.63·22-s + 0.208·23-s − 0.652·24-s − 0.865·25-s + 0.834·26-s − 0.794·27-s + 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 3.40e9T \) |
| good | 2 | \( 1 + 306.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 1.66e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 6.40e4T + 3.05e10T^{2} \) |
| 7 | \( 1 - 1.64e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 6.23e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.11e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 3.06e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 2.24e8T + 1.51e19T^{2} \) |
| 29 | \( 1 + 2.73e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 3.88e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 4.55e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 2.82e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 3.17e11T + 3.17e24T^{2} \) |
| 47 | \( 1 - 1.98e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 8.77e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 2.36e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.02e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 4.55e12T + 2.46e27T^{2} \) |
| 71 | \( 1 + 7.67e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 5.16e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 6.31e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 7.31e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 5.88e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 7.19e14T + 6.33e29T^{2} \) |
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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
−13.93244543207186577518878323333, −11.91271617468246295578902892678, −10.60998229374855523188160804313, −9.570556874795493154679483673182, −8.282095500532096726274993317221, −7.55408730196286900718313944884, −5.53172575093071991502074214918, −2.78812037334677016300500667417, −1.53667208215708452878794704713, 0,
1.53667208215708452878794704713, 2.78812037334677016300500667417, 5.53172575093071991502074214918, 7.55408730196286900718313944884, 8.282095500532096726274993317221, 9.570556874795493154679483673182, 10.60998229374855523188160804313, 11.91271617468246295578902892678, 13.93244543207186577518878323333
