| L(s) = 1 | + 2-s + 2.86·3-s + 4-s − 5-s + 2.86·6-s + 2.76·7-s + 8-s + 5.19·9-s − 10-s + 11-s + 2.86·12-s − 2.67·13-s + 2.76·14-s − 2.86·15-s + 16-s − 2.89·17-s + 5.19·18-s + 1.41·19-s − 20-s + 7.90·21-s + 22-s + 6.20·23-s + 2.86·24-s + 25-s − 2.67·26-s + 6.29·27-s + 2.76·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.65·3-s + 0.5·4-s − 0.447·5-s + 1.16·6-s + 1.04·7-s + 0.353·8-s + 1.73·9-s − 0.316·10-s + 0.301·11-s + 0.826·12-s − 0.741·13-s + 0.738·14-s − 0.739·15-s + 0.250·16-s − 0.702·17-s + 1.22·18-s + 0.325·19-s − 0.223·20-s + 1.72·21-s + 0.213·22-s + 1.29·23-s + 0.584·24-s + 0.200·25-s − 0.524·26-s + 1.21·27-s + 0.521·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.863049328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.863049328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 6.20T + 23T^{2} \) |
| 29 | \( 1 + 1.29T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 + 8.25T + 37T^{2} \) |
| 41 | \( 1 + 0.280T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 8.36T + 59T^{2} \) |
| 61 | \( 1 + 6.83T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 79 | \( 1 + 2.15T + 79T^{2} \) |
| 83 | \( 1 - 4.18T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 2.68T + 97T^{2} \) |
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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
−7.77806870169861387918051406425, −7.27243694536058959045060036673, −6.75044247575835139508070149699, −5.51538669758683907066523633143, −4.74057969411441094999108132286, −4.24610653210728759865216399412, −3.49715979071825312974341282907, −2.71063480513163693144156283049, −2.14031835540895068019133912330, −1.16181769260823480344395365540,
1.16181769260823480344395365540, 2.14031835540895068019133912330, 2.71063480513163693144156283049, 3.49715979071825312974341282907, 4.24610653210728759865216399412, 4.74057969411441094999108132286, 5.51538669758683907066523633143, 6.75044247575835139508070149699, 7.27243694536058959045060036673, 7.77806870169861387918051406425
