| L(s) = 1 | − 1.62·2-s + 0.786·3-s + 0.655·4-s + 5-s − 1.28·6-s + 2.19·8-s − 2.38·9-s − 1.62·10-s + 2.85·11-s + 0.515·12-s − 3.28·13-s + 0.786·15-s − 4.88·16-s + 4.78·17-s + 3.87·18-s − 2.67·19-s + 0.655·20-s − 4.65·22-s + 2.62·23-s + 1.72·24-s + 25-s + 5.35·26-s − 4.23·27-s + 0.965·29-s − 1.28·30-s + 2.64·31-s + 3.57·32-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.454·3-s + 0.327·4-s + 0.447·5-s − 0.523·6-s + 0.774·8-s − 0.793·9-s − 0.515·10-s + 0.861·11-s + 0.148·12-s − 0.912·13-s + 0.203·15-s − 1.22·16-s + 1.16·17-s + 0.914·18-s − 0.613·19-s + 0.146·20-s − 0.992·22-s + 0.547·23-s + 0.351·24-s + 0.200·25-s + 1.05·26-s − 0.814·27-s + 0.179·29-s − 0.234·30-s + 0.475·31-s + 0.631·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 1.62T + 2T^{2} \) |
| 3 | \( 1 - 0.786T + 3T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 - 0.965T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 4.21T + 43T^{2} \) |
| 47 | \( 1 + 8.43T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 - 7.89T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 + 3.11T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.32T + 83T^{2} \) |
| 89 | \( 1 - 5.35T + 89T^{2} \) |
| 97 | \( 1 + 5.11T + 97T^{2} \) |
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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
−7.61165199031268850931849592713, −6.89566220051243292266263693900, −6.23765635951370644293754104040, −5.29704685022406180311011242214, −4.69219426818055998876196057827, −3.66826369856273570858767011473, −2.86511042638927222989909954401, −1.98165899804819769039305854928, −1.17591318669493316178497818846, 0,
1.17591318669493316178497818846, 1.98165899804819769039305854928, 2.86511042638927222989909954401, 3.66826369856273570858767011473, 4.69219426818055998876196057827, 5.29704685022406180311011242214, 6.23765635951370644293754104040, 6.89566220051243292266263693900, 7.61165199031268850931849592713
