L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 2·13-s + 14-s + 16-s − 4·19-s − 20-s + 23-s + 25-s + 2·26-s + 28-s − 4·31-s + 32-s − 35-s + 2·37-s − 4·38-s − 40-s − 12·41-s − 4·43-s + 46-s − 12·47-s + 49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 0.208·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.718·31-s + 0.176·32-s − 0.169·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 1.87·41-s − 0.609·43-s + 0.147·46-s − 1.75·47-s + 1/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.42259932777921, −15.67217678442771, −15.12811103157805, −14.81473872153861, −14.19459939352097, −13.49826160112518, −13.11366362517170, −12.47156779905833, −11.93644923389275, −11.26616360426692, −10.97434473159672, −10.26763730726516, −9.587989066664940, −8.724030369114870, −8.235157376957419, −7.715596135306754, −6.705933503623747, −6.593642401008249, −5.563551150192077, −5.032376704866590, −4.345080926984466, −3.681912217324908, −3.084730710240168, −2.085145004951509, −1.361686951557828, 0,
1.361686951557828, 2.085145004951509, 3.084730710240168, 3.681912217324908, 4.345080926984466, 5.032376704866590, 5.563551150192077, 6.593642401008249, 6.705933503623747, 7.715596135306754, 8.235157376957419, 8.724030369114870, 9.587989066664940, 10.26763730726516, 10.97434473159672, 11.26616360426692, 11.93644923389275, 12.47156779905833, 13.11366362517170, 13.49826160112518, 14.19459939352097, 14.81473872153861, 15.12811103157805, 15.67217678442771, 16.42259932777921