| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 5·19-s − 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-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 + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3126348378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3126348378\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.93356675399703, −13.39746323979951, −12.88321334423791, −12.29758056734258, −11.97764759426959, −11.34436637341654, −11.01438126067602, −10.48561028161175, −10.00943644731680, −9.436815774983529, −8.987811861501665, −8.449882631006038, −7.796730453067150, −7.355803530554789, −6.878574201791485, −6.406049582268974, −5.618133758067181, −5.299848153297241, −4.627457635830215, −3.746105102078350, −3.381432656919009, −2.664949235627336, −1.761220712409575, −1.207554275468046, −0.2251894697401192,
0.2251894697401192, 1.207554275468046, 1.761220712409575, 2.664949235627336, 3.381432656919009, 3.746105102078350, 4.627457635830215, 5.299848153297241, 5.618133758067181, 6.406049582268974, 6.878574201791485, 7.355803530554789, 7.796730453067150, 8.449882631006038, 8.987811861501665, 9.436815774983529, 10.00943644731680, 10.48561028161175, 11.01438126067602, 11.34436637341654, 11.97764759426959, 12.29758056734258, 12.88321334423791, 13.39746323979951, 13.93356675399703
