L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s + 7-s − 3·8-s + 9-s − 10-s − 11-s + 12-s − 2·13-s + 14-s + 15-s − 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 21-s − 22-s − 4·23-s + 3·24-s + 25-s − 2·26-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 − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.213·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.407642504\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.407642504\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.878302477895902313740754811172, −9.044804343455518464188334145368, −7.956661423012176820739921516805, −7.40261507539691697153343679013, −6.09232297254895485500158695203, −5.42282717524508748698587805509, −4.68505520173698243065723732747, −3.83897686808136874613589396246, −2.76907567189605647602440798368, −0.845441805522787961693502909923,
0.845441805522787961693502909923, 2.76907567189605647602440798368, 3.83897686808136874613589396246, 4.68505520173698243065723732747, 5.42282717524508748698587805509, 6.09232297254895485500158695203, 7.40261507539691697153343679013, 7.956661423012176820739921516805, 9.044804343455518464188334145368, 9.878302477895902313740754811172