L(s) = 1 | − 4·2-s − 13·3-s + 16·4-s + 52·6-s + 49·7-s − 64·8-s − 74·9-s − 175·11-s − 208·12-s + 999·13-s − 196·14-s + 256·16-s − 1.83e3·17-s + 296·18-s − 1.30e3·19-s − 637·21-s + 700·22-s + 4.19e3·23-s + 832·24-s − 3.99e3·26-s + 4.12e3·27-s + 784·28-s − 981·29-s − 4.51e3·31-s − 1.02e3·32-s + 2.27e3·33-s + 7.32e3·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.833·3-s + 1/2·4-s + 0.589·6-s + 0.377·7-s − 0.353·8-s − 0.304·9-s − 0.436·11-s − 0.416·12-s + 1.63·13-s − 0.267·14-s + 1/4·16-s − 1.53·17-s + 0.215·18-s − 0.831·19-s − 0.315·21-s + 0.308·22-s + 1.65·23-s + 0.294·24-s − 1.15·26-s + 1.08·27-s + 0.188·28-s − 0.216·29-s − 0.843·31-s − 0.176·32-s + 0.363·33-s + 1.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
good | 3 | \( 1 + 13 T + p^{5} T^{2} \) |
| 11 | \( 1 + 175 T + p^{5} T^{2} \) |
| 13 | \( 1 - 999 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1831 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1308 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4190 T + p^{5} T^{2} \) |
| 29 | \( 1 + 981 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4514 T + p^{5} T^{2} \) |
| 37 | \( 1 - 578 T + p^{5} T^{2} \) |
| 41 | \( 1 - 19526 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10288 T + p^{5} T^{2} \) |
| 47 | \( 1 - 25687 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29874 T + p^{5} T^{2} \) |
| 59 | \( 1 - 1354 T + p^{5} T^{2} \) |
| 61 | \( 1 + 13012 T + p^{5} T^{2} \) |
| 67 | \( 1 + 33026 T + p^{5} T^{2} \) |
| 71 | \( 1 + 21960 T + p^{5} T^{2} \) |
| 73 | \( 1 + 83782 T + p^{5} T^{2} \) |
| 79 | \( 1 + 6417 T + p^{5} T^{2} \) |
| 83 | \( 1 + 7324 T + p^{5} T^{2} \) |
| 89 | \( 1 + 80836 T + p^{5} T^{2} \) |
| 97 | \( 1 + 78575 T + p^{5} T^{2} \) |
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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
−10.68439823149337046596428724618, −8.923831965077257375921836382714, −8.672027823984106731171339168390, −7.29311226903371906652096172428, −6.32792479818803814540149754678, −5.52037151405817253963914537144, −4.21847057302406622961451690343, −2.61695110641039156877294561126, −1.18596983371110833202341530612, 0,
1.18596983371110833202341530612, 2.61695110641039156877294561126, 4.21847057302406622961451690343, 5.52037151405817253963914537144, 6.32792479818803814540149754678, 7.29311226903371906652096172428, 8.672027823984106731171339168390, 8.923831965077257375921836382714, 10.68439823149337046596428724618