L(s) = 1 | − 1.90·2-s + 3-s + 1.62·4-s + 5-s − 1.90·6-s + 4.42·7-s + 0.719·8-s + 9-s − 1.90·10-s + 1.62·12-s + 0.622·13-s − 8.42·14-s + 15-s − 4.61·16-s + 5.18·17-s − 1.90·18-s − 7.05·19-s + 1.62·20-s + 4.42·21-s + 8.85·23-s + 0.719·24-s + 25-s − 1.18·26-s + 27-s + 7.18·28-s + 7.80·29-s − 1.90·30-s + ⋯ |
L(s) = 1 | − 1.34·2-s + 0.577·3-s + 0.811·4-s + 0.447·5-s − 0.776·6-s + 1.67·7-s + 0.254·8-s + 0.333·9-s − 0.601·10-s + 0.468·12-s + 0.172·13-s − 2.25·14-s + 0.258·15-s − 1.15·16-s + 1.25·17-s − 0.448·18-s − 1.61·19-s + 0.362·20-s + 0.966·21-s + 1.84·23-s + 0.146·24-s + 0.200·25-s − 0.232·26-s + 0.192·27-s + 1.35·28-s + 1.44·29-s − 0.347·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.526120102\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526120102\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 - 8.85T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 0.193T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 + 2.75T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 4.85T + 59T^{2} \) |
| 61 | \( 1 + 6.85T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 + 0.133T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 - 7.24T + 97T^{2} \) |
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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
−9.046353764589079796367475493824, −8.466618074808178333124435411419, −8.064980757721626144435657038888, −7.27459501172959312068476464470, −6.37803453785360511975715515429, −5.01621108430893583738914439387, −4.49712345373581957621470083778, −2.95190567342759080754221366648, −1.81019927113047872090365336656, −1.14030064010705934337546709362,
1.14030064010705934337546709362, 1.81019927113047872090365336656, 2.95190567342759080754221366648, 4.49712345373581957621470083778, 5.01621108430893583738914439387, 6.37803453785360511975715515429, 7.27459501172959312068476464470, 8.064980757721626144435657038888, 8.466618074808178333124435411419, 9.046353764589079796367475493824