L(s) = 1 | − 2.08·2-s − 3.08·3-s + 2.35·4-s − 5-s + 6.43·6-s + 1.35·7-s − 0.734·8-s + 6.52·9-s + 2.08·10-s + 3.73·11-s − 7.25·12-s − 2.82·14-s + 3.08·15-s − 3.17·16-s − 2.70·17-s − 13.6·18-s − 0.438·19-s − 2.35·20-s − 4.17·21-s − 7.79·22-s − 5.08·23-s + 2.26·24-s + 25-s − 10.8·27-s + 3.17·28-s − 1.35·29-s − 6.43·30-s + ⋯ |
L(s) = 1 | − 1.47·2-s − 1.78·3-s + 1.17·4-s − 0.447·5-s + 2.62·6-s + 0.510·7-s − 0.259·8-s + 2.17·9-s + 0.659·10-s + 1.12·11-s − 2.09·12-s − 0.753·14-s + 0.796·15-s − 0.793·16-s − 0.655·17-s − 3.20·18-s − 0.100·19-s − 0.525·20-s − 0.910·21-s − 1.66·22-s − 1.06·23-s + 0.462·24-s + 0.200·25-s − 2.09·27-s + 0.600·28-s − 0.251·29-s − 1.17·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3359623577\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3359623577\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 3 | \( 1 + 3.08T + 3T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 0.438T + 19T^{2} \) |
| 23 | \( 1 + 5.08T + 23T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 - 7.35T + 37T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 + 0.209T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 + 1.46T + 53T^{2} \) |
| 59 | \( 1 + 2.26T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 0.438T + 71T^{2} \) |
| 73 | \( 1 + 3.69T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 0.475T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 3.29T + 97T^{2} \) |
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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
−10.19600204325659389731516666872, −9.561066085243332294991626983494, −8.502617608257201077844285342872, −7.69282990343602837154133653161, −6.72099056592991964008876784205, −6.23510897675154975372058308926, −4.88935285164555879898381281500, −4.12880889086125541536084173522, −1.77204768468431448219509923102, −0.66007084128403449122492287750,
0.66007084128403449122492287750, 1.77204768468431448219509923102, 4.12880889086125541536084173522, 4.88935285164555879898381281500, 6.23510897675154975372058308926, 6.72099056592991964008876784205, 7.69282990343602837154133653161, 8.502617608257201077844285342872, 9.561066085243332294991626983494, 10.19600204325659389731516666872