| L(s) = 1 | − 1.10·2-s + 3·3-s − 6.78·4-s − 5·5-s − 3.30·6-s + 1.72·7-s + 16.3·8-s + 9·9-s + 5.51·10-s + 30.1·11-s − 20.3·12-s − 36.1·13-s − 1.90·14-s − 15·15-s + 36.2·16-s − 107.·17-s − 9.92·18-s + 131.·19-s + 33.9·20-s + 5.17·21-s − 33.2·22-s + 35.9·23-s + 48.9·24-s + 25·25-s + 39.8·26-s + 27·27-s − 11.7·28-s + ⋯ |
| L(s) = 1 | − 0.389·2-s + 0.577·3-s − 0.847·4-s − 0.447·5-s − 0.225·6-s + 0.0931·7-s + 0.720·8-s + 0.333·9-s + 0.174·10-s + 0.825·11-s − 0.489·12-s − 0.771·13-s − 0.0363·14-s − 0.258·15-s + 0.566·16-s − 1.53·17-s − 0.129·18-s + 1.58·19-s + 0.379·20-s + 0.0537·21-s − 0.321·22-s + 0.326·23-s + 0.416·24-s + 0.200·25-s + 0.300·26-s + 0.192·27-s − 0.0789·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 1.10T + 8T^{2} \) |
| 7 | \( 1 - 1.72T + 343T^{2} \) |
| 11 | \( 1 - 30.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.9T + 1.21e4T^{2} \) |
| 31 | \( 1 + 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 211.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 508.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 238.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 434.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 246.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 242.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 368.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 480.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 680.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−9.944806173852977476673219196568, −9.269880568767055288078240735565, −8.624233238546395170637982759081, −7.63870569348678702859285135695, −6.86383879056277532330549401137, −5.20073945407256993074817547396, −4.30446393105340264804911241381, −3.27023897586208479596273479086, −1.57889007058541081069697262654, 0,
1.57889007058541081069697262654, 3.27023897586208479596273479086, 4.30446393105340264804911241381, 5.20073945407256993074817547396, 6.86383879056277532330549401137, 7.63870569348678702859285135695, 8.624233238546395170637982759081, 9.269880568767055288078240735565, 9.944806173852977476673219196568
