| L(s) = 1 | + 181.·2-s − 6.41e3·3-s − 3.24e4·4-s − 1.16e6·6-s − 1.78e7·8-s − 1.89e6·9-s + 2.08e8·12-s − 1.51e9·13-s − 1.11e9·16-s − 3.44e8·18-s + 7.83e10·23-s + 1.14e11·24-s + 1.52e11·25-s − 2.75e11·26-s + 2.88e11·27-s + 9.62e11·29-s − 1.40e12·31-s + 9.64e11·32-s + 6.14e10·36-s + 9.70e12·39-s − 1.56e13·41-s + 1.42e13·46-s + 4.39e13·47-s + 7.16e12·48-s + 3.32e13·49-s + 2.77e13·50-s + 4.90e13·52-s + ⋯ |
| L(s) = 1 | + 0.710·2-s − 0.977·3-s − 0.494·4-s − 0.694·6-s − 1.06·8-s − 0.0440·9-s + 0.483·12-s − 1.85·13-s − 0.260·16-s − 0.0312·18-s + 23-s + 1.03·24-s + 25-s − 1.31·26-s + 1.02·27-s + 1.92·29-s − 1.64·31-s + 0.877·32-s + 0.0217·36-s + 1.81·39-s − 1.96·41-s + 0.710·46-s + 1.84·47-s + 0.254·48-s + 49-s + 0.710·50-s + 0.917·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(0.9836774203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9836774203\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 7.83e10T \) |
| good | 2 | \( 1 - 181.T + 6.55e4T^{2} \) |
| 3 | \( 1 + 6.41e3T + 4.30e7T^{2} \) |
| 5 | \( 1 - 1.52e11T^{2} \) |
| 7 | \( 1 - 3.32e13T^{2} \) |
| 11 | \( 1 - 4.59e16T^{2} \) |
| 13 | \( 1 + 1.51e9T + 6.65e17T^{2} \) |
| 17 | \( 1 - 4.86e19T^{2} \) |
| 19 | \( 1 - 2.88e20T^{2} \) |
| 29 | \( 1 - 9.62e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + 1.40e12T + 7.27e23T^{2} \) |
| 37 | \( 1 - 1.23e25T^{2} \) |
| 41 | \( 1 + 1.56e13T + 6.37e25T^{2} \) |
| 43 | \( 1 - 1.36e26T^{2} \) |
| 47 | \( 1 - 4.39e13T + 5.66e26T^{2} \) |
| 53 | \( 1 - 3.87e27T^{2} \) |
| 59 | \( 1 + 6.02e13T + 2.15e28T^{2} \) |
| 61 | \( 1 - 3.67e28T^{2} \) |
| 67 | \( 1 - 1.64e29T^{2} \) |
| 71 | \( 1 - 7.09e14T + 4.16e29T^{2} \) |
| 73 | \( 1 + 1.61e15T + 6.50e29T^{2} \) |
| 79 | \( 1 - 2.30e30T^{2} \) |
| 83 | \( 1 - 5.07e30T^{2} \) |
| 89 | \( 1 - 1.54e31T^{2} \) |
| 97 | \( 1 - 6.14e31T^{2} \) |
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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
−14.11073619446266099166336024085, −12.62845953823184035660978521537, −11.91717004391944166157180442578, −10.38729476242867397762562037715, −8.897404913890004174101544798843, −6.92582463672930799993023009882, −5.42378459585019875081667253445, −4.68341819350319973420636801256, −2.86621861419539886578318849481, −0.54528391016935919998093648642,
0.54528391016935919998093648642, 2.86621861419539886578318849481, 4.68341819350319973420636801256, 5.42378459585019875081667253445, 6.92582463672930799993023009882, 8.897404913890004174101544798843, 10.38729476242867397762562037715, 11.91717004391944166157180442578, 12.62845953823184035660978521537, 14.11073619446266099166336024085
