L(s) = 1 | − 2-s − 0.414·3-s + 4-s + 5-s + 0.414·6-s − 1.71·7-s − 8-s − 2.82·9-s − 10-s − 0.378·11-s − 0.414·12-s − 3.03·13-s + 1.71·14-s − 0.414·15-s + 16-s − 0.663·17-s + 2.82·18-s − 3.53·19-s + 20-s + 0.711·21-s + 0.378·22-s + 0.414·24-s + 25-s + 3.03·26-s + 2.41·27-s − 1.71·28-s − 10.0·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.239·3-s + 0.5·4-s + 0.447·5-s + 0.169·6-s − 0.649·7-s − 0.353·8-s − 0.942·9-s − 0.316·10-s − 0.114·11-s − 0.119·12-s − 0.841·13-s + 0.459·14-s − 0.106·15-s + 0.250·16-s − 0.161·17-s + 0.666·18-s − 0.810·19-s + 0.223·20-s + 0.155·21-s + 0.0807·22-s + 0.0845·24-s + 0.200·25-s + 0.595·26-s + 0.464·27-s − 0.324·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6170088216\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6170088216\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 + 0.378T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 + 0.663T + 17T^{2} \) |
| 19 | \( 1 + 3.53T + 19T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 + 0.828T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 + 8.00T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 4.09T + 61T^{2} \) |
| 67 | \( 1 - 2.67T + 67T^{2} \) |
| 71 | \( 1 - 2.21T + 71T^{2} \) |
| 73 | \( 1 - 2.25T + 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 - 2.40T + 83T^{2} \) |
| 89 | \( 1 - 9.22T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.261853431130462143501595451057, −7.55335715920696084498347922681, −6.68571756782290820368798190723, −6.18764041742183872613616707016, −5.48099275268726156724965373743, −4.64906248788564674719244370500, −3.46952066519259627448587940739, −2.65407020754443730282831664399, −1.91967361735055647142871160617, −0.45010013901976643370175544851,
0.45010013901976643370175544851, 1.91967361735055647142871160617, 2.65407020754443730282831664399, 3.46952066519259627448587940739, 4.64906248788564674719244370500, 5.48099275268726156724965373743, 6.18764041742183872613616707016, 6.68571756782290820368798190723, 7.55335715920696084498347922681, 8.261853431130462143501595451057