| L(s) = 1 | − 3.03·3-s + 5-s − 1.69·7-s + 6.23·9-s + 6.10·11-s − 5.68·13-s − 3.03·15-s + 3.98·17-s + 7.23·19-s + 5.13·21-s + 0.0633·23-s + 25-s − 9.83·27-s + 6.97·29-s − 9.35·31-s − 18.5·33-s − 1.69·35-s + 11.3·37-s + 17.2·39-s + 10.8·41-s + 6.11·43-s + 6.23·45-s − 8.87·47-s − 4.13·49-s − 12.1·51-s + 6.52·53-s + 6.10·55-s + ⋯ |
| L(s) = 1 | − 1.75·3-s + 0.447·5-s − 0.639·7-s + 2.07·9-s + 1.84·11-s − 1.57·13-s − 0.784·15-s + 0.965·17-s + 1.65·19-s + 1.12·21-s + 0.0132·23-s + 0.200·25-s − 1.89·27-s + 1.29·29-s − 1.67·31-s − 3.22·33-s − 0.285·35-s + 1.86·37-s + 2.76·39-s + 1.69·41-s + 0.932·43-s + 0.929·45-s − 1.29·47-s − 0.591·49-s − 1.69·51-s + 0.895·53-s + 0.822·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.239044048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239044048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 - 3.98T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 0.0633T + 23T^{2} \) |
| 29 | \( 1 - 6.97T + 29T^{2} \) |
| 31 | \( 1 + 9.35T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 - 6.52T + 53T^{2} \) |
| 59 | \( 1 + 6.12T + 59T^{2} \) |
| 61 | \( 1 - 0.903T + 61T^{2} \) |
| 67 | \( 1 + 7.35T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 - 5.80T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 4.61T + 83T^{2} \) |
| 89 | \( 1 - 7.00T + 89T^{2} \) |
| 97 | \( 1 + 7.77T + 97T^{2} \) |
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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
−7.63109967767332542858784186220, −7.24101290449485823329472094980, −6.43906588253239319183057947754, −6.01353570266718536220752747119, −5.32064319350230240163917947537, −4.66689855778605146802965772116, −3.84731385257057042101021308352, −2.79748344664871594651713231582, −1.40758519578981287267434499955, −0.71822267237707548885811824679,
0.71822267237707548885811824679, 1.40758519578981287267434499955, 2.79748344664871594651713231582, 3.84731385257057042101021308352, 4.66689855778605146802965772116, 5.32064319350230240163917947537, 6.01353570266718536220752747119, 6.43906588253239319183057947754, 7.24101290449485823329472094980, 7.63109967767332542858784186220
