| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s − 2·9-s + 10-s + 3·11-s + 12-s + 4·13-s + 14-s + 15-s + 16-s − 6·17-s − 2·18-s + 20-s + 21-s + 3·22-s + 6·23-s + 24-s + 25-s + 4·26-s − 5·27-s + 28-s + 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.471·18-s + 0.223·20-s + 0.218·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.962·27-s + 0.188·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.154922010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.154922010\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.73419874569855, −13.34443642737169, −13.07135398254662, −12.28373162107120, −11.87694398011219, −11.17988868391088, −10.91638604851543, −10.67985229516301, −9.568007469818122, −9.215089868787024, −8.825790029517254, −8.367285186909627, −7.767813739028613, −7.054876537673783, −6.533567425524535, −6.228473619442083, −5.467691724302790, −5.127608260796243, −4.258052611607661, −3.927598683907679, −3.311988096050527, −2.599989308009612, −2.180038751762173, −1.437856372192879, −0.7243495325707646,
0.7243495325707646, 1.437856372192879, 2.180038751762173, 2.599989308009612, 3.311988096050527, 3.927598683907679, 4.258052611607661, 5.127608260796243, 5.467691724302790, 6.228473619442083, 6.533567425524535, 7.054876537673783, 7.767813739028613, 8.367285186909627, 8.825790029517254, 9.215089868787024, 9.568007469818122, 10.67985229516301, 10.91638604851543, 11.17988868391088, 11.87694398011219, 12.28373162107120, 13.07135398254662, 13.34443642737169, 13.73419874569855
