L(s) = 1 | + 3.30·3-s + 3.30·5-s + 7.90·9-s + 11-s + 5·13-s + 10.9·15-s − 6.21·17-s − 3.90·19-s − 3.30·23-s + 5.90·25-s + 16.2·27-s + 10.3·29-s − 0.0916·31-s + 3.30·33-s − 0.605·37-s + 16.5·39-s + 41-s − 11.6·43-s + 26.1·45-s + 5.39·47-s − 20.5·51-s − 2.69·53-s + 3.30·55-s − 12.9·57-s − 5.90·59-s + 10.2·61-s + 16.5·65-s + ⋯ |
L(s) = 1 | + 1.90·3-s + 1.47·5-s + 2.63·9-s + 0.301·11-s + 1.38·13-s + 2.81·15-s − 1.50·17-s − 0.896·19-s − 0.688·23-s + 1.18·25-s + 3.11·27-s + 1.91·29-s − 0.0164·31-s + 0.574·33-s − 0.0995·37-s + 2.64·39-s + 0.156·41-s − 1.76·43-s + 3.89·45-s + 0.786·47-s − 2.87·51-s − 0.370·53-s + 0.445·55-s − 1.70·57-s − 0.769·59-s + 1.30·61-s + 2.04·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.466185138\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.466185138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 6.21T + 17T^{2} \) |
| 19 | \( 1 + 3.90T + 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 0.0916T + 31T^{2} \) |
| 37 | \( 1 + 0.605T + 37T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 5.39T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 5.90T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 8.51T + 89T^{2} \) |
| 97 | \( 1 + 9.11T + 97T^{2} \) |
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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
−8.188321110831117079718253589436, −7.09821661694762481092699688293, −6.46972429020839511798112126250, −6.06424743798278492719366442306, −4.75066671987062043245986007123, −4.18394956186334251089442632284, −3.34228521515358622384660519204, −2.53635105582756006745090762891, −1.96087733388379321563108560913, −1.31086234224679349344492469723,
1.31086234224679349344492469723, 1.96087733388379321563108560913, 2.53635105582756006745090762891, 3.34228521515358622384660519204, 4.18394956186334251089442632284, 4.75066671987062043245986007123, 6.06424743798278492719366442306, 6.46972429020839511798112126250, 7.09821661694762481092699688293, 8.188321110831117079718253589436