| L(s) = 1 | − 2-s − 3-s + 4-s + 1.81·5-s + 6-s − 8-s + 9-s − 1.81·10-s + 2.47·11-s − 12-s + 4.37·13-s − 1.81·15-s + 16-s + 7.12·17-s − 18-s + 19-s + 1.81·20-s − 2.47·22-s + 7.89·23-s + 24-s − 1.71·25-s − 4.37·26-s − 27-s + 6.29·29-s + 1.81·30-s + 4.63·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.810·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.572·10-s + 0.745·11-s − 0.288·12-s + 1.21·13-s − 0.467·15-s + 0.250·16-s + 1.72·17-s − 0.235·18-s + 0.229·19-s + 0.405·20-s − 0.527·22-s + 1.64·23-s + 0.204·24-s − 0.343·25-s − 0.857·26-s − 0.192·27-s + 1.16·29-s + 0.330·30-s + 0.833·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911136877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911136877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.81T + 5T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 - 4.79T + 37T^{2} \) |
| 41 | \( 1 + 2.35T + 41T^{2} \) |
| 43 | \( 1 - 1.89T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 8.55T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 3.59T + 67T^{2} \) |
| 71 | \( 1 - 4.79T + 71T^{2} \) |
| 73 | \( 1 - 7.30T + 73T^{2} \) |
| 79 | \( 1 - 5.44T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 4.88T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−8.165264738272299808322309100535, −7.46707106141729236822030255617, −6.58822416692412762516869251533, −6.10954394962854678018002812760, −5.52890579463591333503630278823, −4.60901056188071057668907398077, −3.52617595960885969539205657715, −2.73747737176859734305625025744, −1.33725127417469044667545215767, −1.04853234748252422548936104631,
1.04853234748252422548936104631, 1.33725127417469044667545215767, 2.73747737176859734305625025744, 3.52617595960885969539205657715, 4.60901056188071057668907398077, 5.52890579463591333503630278823, 6.10954394962854678018002812760, 6.58822416692412762516869251533, 7.46707106141729236822030255617, 8.165264738272299808322309100535
