| L(s) = 1 | + 2-s − 0.603·3-s + 4-s + 5-s − 0.603·6-s + 3.60·7-s + 8-s − 2.63·9-s + 10-s + 3.83·11-s − 0.603·12-s + 0.122·13-s + 3.60·14-s − 0.603·15-s + 16-s − 2.31·17-s − 2.63·18-s − 6.64·19-s + 20-s − 2.17·21-s + 3.83·22-s − 4.09·23-s − 0.603·24-s + 25-s + 0.122·26-s + 3.40·27-s + 3.60·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.348·3-s + 0.5·4-s + 0.447·5-s − 0.246·6-s + 1.36·7-s + 0.353·8-s − 0.878·9-s + 0.316·10-s + 1.15·11-s − 0.174·12-s + 0.0340·13-s + 0.963·14-s − 0.155·15-s + 0.250·16-s − 0.562·17-s − 0.621·18-s − 1.52·19-s + 0.223·20-s − 0.474·21-s + 0.816·22-s − 0.852·23-s − 0.123·24-s + 0.200·25-s + 0.0240·26-s + 0.654·27-s + 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.801643256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.801643256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 0.603T + 3T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 - 0.122T + 13T^{2} \) |
| 17 | \( 1 + 2.31T + 17T^{2} \) |
| 19 | \( 1 + 6.64T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 31 | \( 1 - 9.44T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 9.28T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 2.73T + 53T^{2} \) |
| 59 | \( 1 + 0.361T + 59T^{2} \) |
| 61 | \( 1 + 6.29T + 61T^{2} \) |
| 67 | \( 1 + 4.42T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 + 2.55T + 89T^{2} \) |
| 97 | \( 1 - 8.66T + 97T^{2} \) |
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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
−7.83472575783775835828635951976, −6.84493629715647397910789316998, −6.08753133258639913725306379836, −5.96992043519554530342107935842, −4.75678010388597597073233167146, −4.54983703799827159368374571818, −3.68754119394793805145091061063, −2.48322307964426296850221947465, −1.97428045413643277384787682671, −0.893788516859927770586249140087,
0.893788516859927770586249140087, 1.97428045413643277384787682671, 2.48322307964426296850221947465, 3.68754119394793805145091061063, 4.54983703799827159368374571818, 4.75678010388597597073233167146, 5.96992043519554530342107935842, 6.08753133258639913725306379836, 6.84493629715647397910789316998, 7.83472575783775835828635951976
