| L(s) = 1 | − 2.47·2-s + 2.38·3-s + 4.12·4-s − 5-s − 5.90·6-s + 4.78·7-s − 5.24·8-s + 2.69·9-s + 2.47·10-s − 3.12·11-s + 9.83·12-s − 2.81·13-s − 11.8·14-s − 2.38·15-s + 4.74·16-s + 6.37·17-s − 6.67·18-s + 0.114·19-s − 4.12·20-s + 11.4·21-s + 7.72·22-s + 5.62·23-s − 12.5·24-s + 25-s + 6.97·26-s − 0.723·27-s + 19.7·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 1.37·3-s + 2.06·4-s − 0.447·5-s − 2.41·6-s + 1.80·7-s − 1.85·8-s + 0.898·9-s + 0.782·10-s − 0.941·11-s + 2.83·12-s − 0.781·13-s − 3.16·14-s − 0.616·15-s + 1.18·16-s + 1.54·17-s − 1.57·18-s + 0.0262·19-s − 0.921·20-s + 2.49·21-s + 1.64·22-s + 1.17·23-s − 2.55·24-s + 0.200·25-s + 1.36·26-s − 0.139·27-s + 3.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8727793161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8727793161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 3 | \( 1 - 2.38T + 3T^{2} \) |
| 7 | \( 1 - 4.78T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 - 0.114T + 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 + 8.57T + 43T^{2} \) |
| 47 | \( 1 - 3.40T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 5.68T + 83T^{2} \) |
| 89 | \( 1 + 9.95T + 89T^{2} \) |
| 97 | \( 1 + 5.62T + 97T^{2} \) |
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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
−12.28948997712310411512652281172, −11.22561153688173908334151274067, −10.40145374283225890791413832887, −9.336751719744917118039868766198, −8.433251260864881784790189302794, −7.79195259043054011452350378523, −7.42788396209690352090502049831, −5.02781149929306498428302155734, −2.95939231606806056940195278431, −1.65120181560874527199921964844,
1.65120181560874527199921964844, 2.95939231606806056940195278431, 5.02781149929306498428302155734, 7.42788396209690352090502049831, 7.79195259043054011452350378523, 8.433251260864881784790189302794, 9.336751719744917118039868766198, 10.40145374283225890791413832887, 11.22561153688173908334151274067, 12.28948997712310411512652281172
