L(s) = 1 | + 2.77·2-s − 3.16·3-s + 5.67·4-s − 5-s − 8.78·6-s − 2.64·7-s + 10.1·8-s + 7.04·9-s − 2.77·10-s − 1.40·11-s − 17.9·12-s − 1.13·13-s − 7.33·14-s + 3.16·15-s + 16.8·16-s − 2.63·17-s + 19.5·18-s + 7.38·19-s − 5.67·20-s + 8.38·21-s − 3.87·22-s + 4.55·23-s − 32.2·24-s + 25-s − 3.14·26-s − 12.8·27-s − 15.0·28-s + ⋯ |
L(s) = 1 | + 1.95·2-s − 1.83·3-s + 2.83·4-s − 0.447·5-s − 3.58·6-s − 1.00·7-s + 3.60·8-s + 2.34·9-s − 0.876·10-s − 0.422·11-s − 5.19·12-s − 0.314·13-s − 1.95·14-s + 0.818·15-s + 4.21·16-s − 0.638·17-s + 4.60·18-s + 1.69·19-s − 1.26·20-s + 1.83·21-s − 0.826·22-s + 0.950·23-s − 6.59·24-s + 0.200·25-s − 0.616·26-s − 2.46·27-s − 2.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.068087612\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.068087612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 3 | \( 1 + 3.16T + 3T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 - 7.38T + 19T^{2} \) |
| 23 | \( 1 - 4.55T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + 1.77T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 - 6.91T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 + 2.06T + 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 - 9.83T + 67T^{2} \) |
| 71 | \( 1 + 4.63T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 9.01T + 97T^{2} \) |
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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
−9.599760490246052084573062379430, −7.68529141022403895238304810763, −7.05628617383008388092874900042, −6.43830817855491170188042924192, −5.87588062813756121810958460648, −4.98807282173240749759530662124, −4.65589107766118451397427290465, −3.57803274155435559554297823135, −2.68645676749179331423325386588, −0.979737891810172423484342146491,
0.979737891810172423484342146491, 2.68645676749179331423325386588, 3.57803274155435559554297823135, 4.65589107766118451397427290465, 4.98807282173240749759530662124, 5.87588062813756121810958460648, 6.43830817855491170188042924192, 7.05628617383008388092874900042, 7.68529141022403895238304810763, 9.599760490246052084573062379430