L(s) = 1 | + 2-s − 3.23·3-s + 4-s + 3.23·5-s − 3.23·6-s + 7-s + 8-s + 7.47·9-s + 3.23·10-s + 11-s − 3.23·12-s + 1.23·13-s + 14-s − 10.4·15-s + 16-s − 6.47·17-s + 7.47·18-s − 2.76·19-s + 3.23·20-s − 3.23·21-s + 22-s + 4·23-s − 3.23·24-s + 5.47·25-s + 1.23·26-s − 14.4·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.86·3-s + 0.5·4-s + 1.44·5-s − 1.32·6-s + 0.377·7-s + 0.353·8-s + 2.49·9-s + 1.02·10-s + 0.301·11-s − 0.934·12-s + 0.342·13-s + 0.267·14-s − 2.70·15-s + 0.250·16-s − 1.56·17-s + 1.76·18-s − 0.634·19-s + 0.723·20-s − 0.706·21-s + 0.213·22-s + 0.834·23-s − 0.660·24-s + 1.09·25-s + 0.242·26-s − 2.78·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.246811670\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246811670\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 4.94T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−13.02165373672923789671526488169, −11.95456775219894293344517668118, −10.97803505558155713549145959874, −10.46813122479265160040897078207, −9.141839153069202927093459272189, −6.94310403447038207708400147483, −6.23746833597972884543119043828, −5.40100649640324540624234594736, −4.45294812940400001555688902048, −1.77756563632127092242172589873,
1.77756563632127092242172589873, 4.45294812940400001555688902048, 5.40100649640324540624234594736, 6.23746833597972884543119043828, 6.94310403447038207708400147483, 9.141839153069202927093459272189, 10.46813122479265160040897078207, 10.97803505558155713549145959874, 11.95456775219894293344517668118, 13.02165373672923789671526488169