L(s) = 1 | − 0.732i·3-s + 1.26·7-s + 2.46·9-s + 3.46i·11-s − 3.46i·13-s − 3.46·17-s + 2i·19-s − 0.928i·21-s + 8.19·23-s − 4i·27-s + 9.46·31-s + 2.53·33-s + 6i·37-s − 2.53·39-s + 2.53·41-s + ⋯ |
L(s) = 1 | − 0.422i·3-s + 0.479·7-s + 0.821·9-s + 1.04i·11-s − 0.960i·13-s − 0.840·17-s + 0.458i·19-s − 0.202i·21-s + 1.70·23-s − 0.769i·27-s + 1.69·31-s + 0.441·33-s + 0.986i·37-s − 0.406·39-s + 0.396·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.976115583\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976115583\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 0.732iT - 3T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 - 10.1iT - 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4.73iT - 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−9.597535507399314581612047420962, −8.281515675094429053237370331237, −7.907599565595621473394738781378, −6.87408489466295983647596598392, −6.43599974479509101738743143551, −4.95285992577437578539031471022, −4.65014674744689760221769797152, −3.28134458687571959580922266287, −2.12734496613596434979142475954, −1.06877578716346748447680640523,
1.04103282447108542597421830449, 2.38341900110084521354369894903, 3.57360831731957384378284412077, 4.51210613575953620226129260921, 5.08293514593266348123808184123, 6.32609026687056091261396070132, 6.96110452104945527557904314030, 7.87745808284788022029322611772, 8.983713761928337028640575515390, 9.130106679394828965763453379568