L(s) = 1 | − 2.47·2-s − 3-s + 4.10·4-s − 0.537·5-s + 2.47·6-s + 7-s − 5.19·8-s + 9-s + 1.32·10-s + 3.20·11-s − 4.10·12-s − 0.0237·13-s − 2.47·14-s + 0.537·15-s + 4.62·16-s − 5.37·17-s − 2.47·18-s + 0.850·19-s − 2.20·20-s − 21-s − 7.91·22-s + 4.95·23-s + 5.19·24-s − 4.71·25-s + 0.0585·26-s − 27-s + 4.10·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s − 0.577·3-s + 2.05·4-s − 0.240·5-s + 1.00·6-s + 0.377·7-s − 1.83·8-s + 0.333·9-s + 0.419·10-s + 0.966·11-s − 1.18·12-s − 0.00657·13-s − 0.660·14-s + 0.138·15-s + 1.15·16-s − 1.30·17-s − 0.582·18-s + 0.195·19-s − 0.492·20-s − 0.218·21-s − 1.68·22-s + 1.03·23-s + 1.06·24-s − 0.942·25-s + 0.0114·26-s − 0.192·27-s + 0.775·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 5 | \( 1 + 0.537T + 5T^{2} \) |
| 11 | \( 1 - 3.20T + 11T^{2} \) |
| 13 | \( 1 + 0.0237T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 0.850T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 - 6.71T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 + 6.22T + 43T^{2} \) |
| 47 | \( 1 - 1.06T + 47T^{2} \) |
| 53 | \( 1 - 7.96T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 9.86T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 - 6.02T + 83T^{2} \) |
| 89 | \( 1 + 3.71T + 89T^{2} \) |
| 97 | \( 1 + 5.84T + 97T^{2} \) |
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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
−7.47210917504057457885128891446, −6.94665522810305886490499008367, −6.52558987803427700852966653134, −5.63658661183662667603249379379, −4.68059669605515221779092409928, −3.90074923856688246491394232676, −2.71660220808005089683851496926, −1.78166413814835139066233295986, −1.04883735517930484589056297955, 0,
1.04883735517930484589056297955, 1.78166413814835139066233295986, 2.71660220808005089683851496926, 3.90074923856688246491394232676, 4.68059669605515221779092409928, 5.63658661183662667603249379379, 6.52558987803427700852966653134, 6.94665522810305886490499008367, 7.47210917504057457885128891446