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
−9.130106679394828965763453379568, −8.983713761928337028640575515390, −7.87745808284788022029322611772, −6.96110452104945527557904314030, −6.32609026687056091261396070132, −5.08293514593266348123808184123, −4.51210613575953620226129260921, −3.57360831731957384378284412077, −2.38341900110084521354369894903, −1.04103282447108542597421830449,
1.06877578716346748447680640523, 2.12734496613596434979142475954, 3.28134458687571959580922266287, 4.65014674744689760221769797152, 4.95285992577437578539031471022, 6.43599974479509101738743143551, 6.87408489466295983647596598392, 7.907599565595621473394738781378, 8.281515675094429053237370331237, 9.597535507399314581612047420962