| L(s) = 1 | − 2-s + 4-s + 0.856·5-s − 2.82·7-s − 8-s − 0.856·10-s − 5.63·11-s + 0.572·13-s + 2.82·14-s + 16-s − 1.82·17-s + 7.67·19-s + 0.856·20-s + 5.63·22-s − 4.26·25-s − 0.572·26-s − 2.82·28-s + 4.97·29-s + 8.12·31-s − 32-s + 1.82·34-s − 2.41·35-s + 0.415·37-s − 7.67·38-s − 0.856·40-s + 2.84·41-s − 5.30·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.383·5-s − 1.06·7-s − 0.353·8-s − 0.270·10-s − 1.70·11-s + 0.158·13-s + 0.754·14-s + 0.250·16-s − 0.442·17-s + 1.76·19-s + 0.191·20-s + 1.20·22-s − 0.853·25-s − 0.112·26-s − 0.533·28-s + 0.923·29-s + 1.45·31-s − 0.176·32-s + 0.312·34-s − 0.408·35-s + 0.0682·37-s − 1.24·38-s − 0.135·40-s + 0.443·41-s − 0.808·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 0.856T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 5.63T + 11T^{2} \) |
| 13 | \( 1 - 0.572T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 - 7.67T + 19T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 - 8.12T + 31T^{2} \) |
| 37 | \( 1 - 0.415T + 37T^{2} \) |
| 41 | \( 1 - 2.84T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 + 0.421T + 53T^{2} \) |
| 59 | \( 1 - 6.90T + 59T^{2} \) |
| 61 | \( 1 + 5.51T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 4.53T + 71T^{2} \) |
| 73 | \( 1 + 2.63T + 73T^{2} \) |
| 79 | \( 1 - 4.59T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 3.19T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.46538931563539803069913099298, −6.69596700440102691957455164497, −6.12827866124208465720277997450, −5.39951356327667412234803439439, −4.75869041074631404539349508926, −3.49937302813551603016826329277, −2.88481833580945853747308474740, −2.27092341515292811521663815937, −1.02811519979872370968970838917, 0,
1.02811519979872370968970838917, 2.27092341515292811521663815937, 2.88481833580945853747308474740, 3.49937302813551603016826329277, 4.75869041074631404539349508926, 5.39951356327667412234803439439, 6.12827866124208465720277997450, 6.69596700440102691957455164497, 7.46538931563539803069913099298
