L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s + 3·11-s − 4·13-s − 14-s + 16-s − 6·17-s − 7·19-s − 3·20-s − 3·22-s − 3·23-s + 4·25-s + 4·26-s + 28-s + 5·31-s − 32-s + 6·34-s − 3·35-s − 7·37-s + 7·38-s + 3·40-s − 9·41-s − 10·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 0.904·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.60·19-s − 0.670·20-s − 0.639·22-s − 0.625·23-s + 4/5·25-s + 0.784·26-s + 0.188·28-s + 0.898·31-s − 0.176·32-s + 1.02·34-s − 0.507·35-s − 1.15·37-s + 1.13·38-s + 0.474·40-s − 1.40·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−10.92490183559974483870177287472, −10.05189465182372194736077139645, −8.751755972478050515328668545232, −8.335493328080316690083698704832, −7.19284214678952744065952854249, −6.52958134533989476002257403650, −4.74118220843440043197796971108, −3.84049559583729782963394147722, −2.14382090414345698958299236646, 0,
2.14382090414345698958299236646, 3.84049559583729782963394147722, 4.74118220843440043197796971108, 6.52958134533989476002257403650, 7.19284214678952744065952854249, 8.335493328080316690083698704832, 8.751755972478050515328668545232, 10.05189465182372194736077139645, 10.92490183559974483870177287472