| L(s) = 1 | + 2.74·2-s − 0.210·3-s + 5.53·4-s − 5-s − 0.578·6-s + 2.32·7-s + 9.70·8-s − 2.95·9-s − 2.74·10-s − 1.16·12-s + 0.534·13-s + 6.37·14-s + 0.210·15-s + 15.5·16-s − 2.42·17-s − 8.11·18-s − 4.95·19-s − 5.53·20-s − 0.489·21-s − 4.53·23-s − 2.04·24-s + 25-s + 1.46·26-s + 1.25·27-s + 12.8·28-s + 5.48·29-s + 0.578·30-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.121·3-s + 2.76·4-s − 0.447·5-s − 0.236·6-s + 0.878·7-s + 3.42·8-s − 0.985·9-s − 0.867·10-s − 0.336·12-s + 0.148·13-s + 1.70·14-s + 0.0544·15-s + 3.88·16-s − 0.587·17-s − 1.91·18-s − 1.13·19-s − 1.23·20-s − 0.106·21-s − 0.945·23-s − 0.417·24-s + 0.200·25-s + 0.287·26-s + 0.241·27-s + 2.42·28-s + 1.01·29-s + 0.105·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.322416584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.322416584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 3 | \( 1 + 0.210T + 3T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 13 | \( 1 - 0.534T + 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 + 4.53T + 23T^{2} \) |
| 29 | \( 1 - 5.48T + 29T^{2} \) |
| 31 | \( 1 - 1.04T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 + 6.79T + 61T^{2} \) |
| 67 | \( 1 - 0.721T + 67T^{2} \) |
| 71 | \( 1 - 4.53T + 71T^{2} \) |
| 73 | \( 1 + 1.06T + 73T^{2} \) |
| 79 | \( 1 + 4.64T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 4.75T + 97T^{2} \) |
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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
−11.11262556002082967296493524052, −10.38018463444711668970205940011, −8.520359067983594798763783476830, −7.81719587086280632313461149505, −6.61055564484694900606052599517, −5.96050896947728302173885870850, −4.87777436584057675282077828439, −4.28157916889313020125857102242, −3.10781789215065694744696590911, −1.99949519016452146689239223328,
1.99949519016452146689239223328, 3.10781789215065694744696590911, 4.28157916889313020125857102242, 4.87777436584057675282077828439, 5.96050896947728302173885870850, 6.61055564484694900606052599517, 7.81719587086280632313461149505, 8.520359067983594798763783476830, 10.38018463444711668970205940011, 11.11262556002082967296493524052
