L(s) = 1 | + 1.38·2-s − 0.0816·4-s + 2.56·5-s + 0.393·7-s − 2.88·8-s + 3.55·10-s − 0.0100·11-s − 1.56·13-s + 0.544·14-s − 3.82·16-s − 0.120·17-s − 19-s − 0.209·20-s − 0.0139·22-s + 3.27·23-s + 1.59·25-s − 2.17·26-s − 0.0321·28-s − 9.33·29-s − 4.52·31-s + 0.461·32-s − 0.167·34-s + 1.00·35-s − 2.88·37-s − 1.38·38-s − 7.40·40-s + 7.99·41-s + ⋯ |
L(s) = 1 | + 0.979·2-s − 0.0408·4-s + 1.14·5-s + 0.148·7-s − 1.01·8-s + 1.12·10-s − 0.00303·11-s − 0.435·13-s + 0.145·14-s − 0.957·16-s − 0.0292·17-s − 0.229·19-s − 0.0468·20-s − 0.00297·22-s + 0.682·23-s + 0.318·25-s − 0.426·26-s − 0.00606·28-s − 1.73·29-s − 0.813·31-s + 0.0816·32-s − 0.0286·34-s + 0.170·35-s − 0.473·37-s − 0.224·38-s − 1.17·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 - 1.38T + 2T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 0.393T + 7T^{2} \) |
| 11 | \( 1 + 0.0100T + 11T^{2} \) |
| 13 | \( 1 + 1.56T + 13T^{2} \) |
| 17 | \( 1 + 0.120T + 17T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 + 9.33T + 29T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 + 2.88T + 37T^{2} \) |
| 41 | \( 1 - 7.99T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 53 | \( 1 + 1.80T + 53T^{2} \) |
| 59 | \( 1 - 8.28T + 59T^{2} \) |
| 61 | \( 1 + 7.98T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 0.336T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 9.82T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.25259195461348344127364402696, −6.61012002161218123560877853085, −5.79997732866965055932939467725, −5.42963471871886082901226272766, −4.79424216650040477333327080655, −3.96832766044526519719050830146, −3.20503803967559737853729619829, −2.35062795429821596571641466441, −1.56511543658422838313663948349, 0,
1.56511543658422838313663948349, 2.35062795429821596571641466441, 3.20503803967559737853729619829, 3.96832766044526519719050830146, 4.79424216650040477333327080655, 5.42963471871886082901226272766, 5.79997732866965055932939467725, 6.61012002161218123560877853085, 7.25259195461348344127364402696