L(s) = 1 | − 2·5-s + 7-s + 11-s − 6·13-s + 5·17-s + 7·19-s + 4·23-s − 25-s + 4·29-s + 6·31-s − 2·35-s + 2·37-s − 3·41-s + 43-s + 49-s − 12·53-s − 2·55-s − 7·59-s − 12·61-s + 12·65-s − 13·67-s − 8·71-s + 73-s + 77-s + 6·79-s + 16·83-s − 10·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.301·11-s − 1.66·13-s + 1.21·17-s + 1.60·19-s + 0.834·23-s − 1/5·25-s + 0.742·29-s + 1.07·31-s − 0.338·35-s + 0.328·37-s − 0.468·41-s + 0.152·43-s + 1/7·49-s − 1.64·53-s − 0.269·55-s − 0.911·59-s − 1.53·61-s + 1.48·65-s − 1.58·67-s − 0.949·71-s + 0.117·73-s + 0.113·77-s + 0.675·79-s + 1.75·83-s − 1.08·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.691990928\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.691990928\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.70052737375427810527867961102, −7.32328142034212169652637712101, −6.45483384618643184396126718601, −5.53922034169113861911000590615, −4.82509621746485422373709388802, −4.42960323670660442130542543389, −3.21260118060593360466087975162, −2.95956006382655888545987880486, −1.60856331080708533555163710204, −0.65079456429759512487140242478,
0.65079456429759512487140242478, 1.60856331080708533555163710204, 2.95956006382655888545987880486, 3.21260118060593360466087975162, 4.42960323670660442130542543389, 4.82509621746485422373709388802, 5.53922034169113861911000590615, 6.45483384618643184396126718601, 7.32328142034212169652637712101, 7.70052737375427810527867961102