| L(s) = 1 | − 1.64·3-s + 1.84·5-s − 3.60·7-s − 0.286·9-s − 11-s − 4.60·13-s − 3.04·15-s + 4.53·17-s − 19-s + 5.93·21-s − 7.43·23-s − 1.58·25-s + 5.41·27-s − 1.20·29-s + 8.16·31-s + 1.64·33-s − 6.66·35-s − 2.67·37-s + 7.58·39-s − 10.6·41-s − 7.58·43-s − 0.530·45-s + 7.21·47-s + 5.99·49-s − 7.47·51-s + 9.86·53-s − 1.84·55-s + ⋯ |
| L(s) = 1 | − 0.950·3-s + 0.826·5-s − 1.36·7-s − 0.0956·9-s − 0.301·11-s − 1.27·13-s − 0.786·15-s + 1.10·17-s − 0.229·19-s + 1.29·21-s − 1.54·23-s − 0.316·25-s + 1.04·27-s − 0.224·29-s + 1.46·31-s + 0.286·33-s − 1.12·35-s − 0.439·37-s + 1.21·39-s − 1.66·41-s − 1.15·43-s − 0.0790·45-s + 1.05·47-s + 0.856·49-s − 1.04·51-s + 1.35·53-s − 0.249·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7133699303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7133699303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 - 8.16T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 7.58T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 9.86T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 9.37T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 8.17T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.666153518473411213788987770975, −7.81988535509263104963795041391, −6.78723241503771360434134089204, −6.38043780685526121434160217439, −5.53181829832949331060511102130, −5.21700012055662789314325447688, −3.94887585684137193636159324073, −2.94504671304132314976762988574, −2.09932245557274451815775135064, −0.49749908859948295515717762693,
0.49749908859948295515717762693, 2.09932245557274451815775135064, 2.94504671304132314976762988574, 3.94887585684137193636159324073, 5.21700012055662789314325447688, 5.53181829832949331060511102130, 6.38043780685526121434160217439, 6.78723241503771360434134089204, 7.81988535509263104963795041391, 8.666153518473411213788987770975
