| L(s) = 1 | − 150.·2-s + 3.98e3·3-s − 1.00e4·4-s + 3.47e5·5-s − 6.00e5·6-s − 3.26e6·7-s + 6.45e6·8-s + 1.53e6·9-s − 5.23e7·10-s − 2.77e7·11-s − 4.01e7·12-s − 3.36e8·13-s + 4.91e8·14-s + 1.38e9·15-s − 6.41e8·16-s − 5.77e8·17-s − 2.31e8·18-s + 2.00e9·19-s − 3.50e9·20-s − 1.30e10·21-s + 4.18e9·22-s + 3.40e9·23-s + 2.57e10·24-s + 9.04e10·25-s + 5.06e10·26-s − 5.10e10·27-s + 3.29e10·28-s + ⋯ |
| L(s) = 1 | − 0.832·2-s + 1.05·3-s − 0.307·4-s + 1.99·5-s − 0.875·6-s − 1.49·7-s + 1.08·8-s + 0.107·9-s − 1.65·10-s − 0.429·11-s − 0.323·12-s − 1.48·13-s + 1.24·14-s + 2.09·15-s − 0.597·16-s − 0.341·17-s − 0.0891·18-s + 0.514·19-s − 0.612·20-s − 1.57·21-s + 0.357·22-s + 0.208·23-s + 1.14·24-s + 2.96·25-s + 1.23·26-s − 0.939·27-s + 0.460·28-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 | \( 1 + 150.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 3.98e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 3.47e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 3.26e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 2.77e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 3.36e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 5.77e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 2.00e9T + 1.51e19T^{2} \) |
| 29 | \( 1 + 1.29e11T + 8.62e21T^{2} \) |
| 31 | \( 1 - 6.03e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 4.00e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 5.39e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 3.54e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 1.77e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 2.01e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 2.80e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 5.22e12T + 6.02e26T^{2} \) |
| 67 | \( 1 + 1.64e12T + 2.46e27T^{2} \) |
| 71 | \( 1 + 4.70e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.27e14T + 8.90e27T^{2} \) |
| 79 | \( 1 - 2.90e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 3.01e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 1.09e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 7.46e14T + 6.33e29T^{2} \) |
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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
−13.53773300489707393976844507033, −13.05257726606059600663371139249, −10.02297854064309165899004206124, −9.708362218945288415552678072406, −8.789309304514693854735713352848, −7.01297018674707746971289902570, −5.35130228534398035864251135475, −2.97041397530628435804988910903, −1.90589232499104398521118079262, 0,
1.90589232499104398521118079262, 2.97041397530628435804988910903, 5.35130228534398035864251135475, 7.01297018674707746971289902570, 8.789309304514693854735713352848, 9.708362218945288415552678072406, 10.02297854064309165899004206124, 13.05257726606059600663371139249, 13.53773300489707393976844507033
