| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 18·7-s − 8·8-s − 10·10-s + 32·11-s − 47·13-s + 36·14-s + 16·16-s − 20·17-s + 36·19-s + 20·20-s − 64·22-s + 23·23-s + 25·25-s + 94·26-s − 72·28-s + 27·29-s − 33·31-s − 32·32-s + 40·34-s − 90·35-s + 56·37-s − 72·38-s − 40·40-s + 157·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.971·7-s − 0.353·8-s − 0.316·10-s + 0.877·11-s − 1.00·13-s + 0.687·14-s + 1/4·16-s − 0.285·17-s + 0.434·19-s + 0.223·20-s − 0.620·22-s + 0.208·23-s + 1/5·25-s + 0.709·26-s − 0.485·28-s + 0.172·29-s − 0.191·31-s − 0.176·32-s + 0.201·34-s − 0.434·35-s + 0.248·37-s − 0.307·38-s − 0.158·40-s + 0.598·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 - p T \) |
| good | 7 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 47 T + p^{3} T^{2} \) |
| 17 | \( 1 + 20 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 29 | \( 1 - 27 T + p^{3} T^{2} \) |
| 31 | \( 1 + 33 T + p^{3} T^{2} \) |
| 37 | \( 1 - 56 T + p^{3} T^{2} \) |
| 41 | \( 1 - 157 T + p^{3} T^{2} \) |
| 43 | \( 1 - 18 T + p^{3} T^{2} \) |
| 47 | \( 1 + 65 T + p^{3} T^{2} \) |
| 53 | \( 1 - 14 T + p^{3} T^{2} \) |
| 59 | \( 1 - 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 552 T + p^{3} T^{2} \) |
| 67 | \( 1 + 156 T + p^{3} T^{2} \) |
| 71 | \( 1 + 699 T + p^{3} T^{2} \) |
| 73 | \( 1 + 609 T + p^{3} T^{2} \) |
| 79 | \( 1 + 644 T + p^{3} T^{2} \) |
| 83 | \( 1 + 512 T + p^{3} T^{2} \) |
| 89 | \( 1 - 102 T + p^{3} T^{2} \) |
| 97 | \( 1 - 578 T + p^{3} T^{2} \) |
| 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
−8.608246176084150298496602396082, −7.49256360394649991840305465151, −6.88452048452857143759825598061, −6.21830338637405348188622606831, −5.35142508102068939697805029405, −4.20349226170209957092510642580, −3.13439089135123642125059341553, −2.29798757226791552332434209526, −1.13345102020249856276648043416, 0,
1.13345102020249856276648043416, 2.29798757226791552332434209526, 3.13439089135123642125059341553, 4.20349226170209957092510642580, 5.35142508102068939697805029405, 6.21830338637405348188622606831, 6.88452048452857143759825598061, 7.49256360394649991840305465151, 8.608246176084150298496602396082
