L(s) = 1 | − 2.49·2-s + 0.530·3-s + 4.23·4-s + 5-s − 1.32·6-s + 3.05·7-s − 5.59·8-s − 2.71·9-s − 2.49·10-s + 3.61·11-s + 2.25·12-s − 0.556·13-s − 7.63·14-s + 0.530·15-s + 5.49·16-s − 3.15·17-s + 6.78·18-s + 7.04·19-s + 4.23·20-s + 1.62·21-s − 9.02·22-s − 2.38·23-s − 2.96·24-s + 25-s + 1.39·26-s − 3.03·27-s + 12.9·28-s + ⋯ |
L(s) = 1 | − 1.76·2-s + 0.306·3-s + 2.11·4-s + 0.447·5-s − 0.541·6-s + 1.15·7-s − 1.97·8-s − 0.906·9-s − 0.789·10-s + 1.08·11-s + 0.649·12-s − 0.154·13-s − 2.04·14-s + 0.137·15-s + 1.37·16-s − 0.765·17-s + 1.60·18-s + 1.61·19-s + 0.947·20-s + 0.354·21-s − 1.92·22-s − 0.498·23-s − 0.605·24-s + 0.200·25-s + 0.272·26-s − 0.584·27-s + 2.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 1607 | \( 1 - T \) |
good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 - 0.530T + 3T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 0.556T + 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 23 | \( 1 + 2.38T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 0.183T + 31T^{2} \) |
| 37 | \( 1 - 7.94T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 5.15T + 43T^{2} \) |
| 47 | \( 1 + 0.982T + 47T^{2} \) |
| 53 | \( 1 - 3.73T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 5.31T + 67T^{2} \) |
| 71 | \( 1 + 4.29T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 - 9.86T + 83T^{2} \) |
| 89 | \( 1 + 4.55T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−7.74855739560969125214546386637, −7.12888670902710513505039377273, −6.32715339745301640627794488001, −5.64030817724906522641057922714, −4.78774714552763450195032370837, −3.62600815254165327256734335547, −2.69887098573496445275308464197, −1.78836075927227759994698608964, −1.36836379185094502647964826285, 0,
1.36836379185094502647964826285, 1.78836075927227759994698608964, 2.69887098573496445275308464197, 3.62600815254165327256734335547, 4.78774714552763450195032370837, 5.64030817724906522641057922714, 6.32715339745301640627794488001, 7.12888670902710513505039377273, 7.74855739560969125214546386637