L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s − 2·11-s + 2·13-s − 4·14-s + 16-s − 20-s − 2·22-s + 6·23-s + 25-s + 2·26-s − 4·28-s + 8·29-s + 31-s + 32-s + 4·35-s − 2·37-s − 40-s − 6·41-s + 4·43-s − 2·44-s + 6·46-s + 12·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.48·29-s + 0.179·31-s + 0.176·32-s + 0.676·35-s − 0.328·37-s − 0.158·40-s − 0.937·41-s + 0.609·43-s − 0.301·44-s + 0.884·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.108301693\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.108301693\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.807539393288834856710357109022, −7.966391282386719564203088881500, −6.99426412218617473422211274605, −6.58246294038986299270139996042, −5.71872634371991770953386119123, −4.90129340528090247076056131033, −3.91854230155308754016351870095, −3.20585163256306441826577970590, −2.52442889579518216391449621291, −0.796197486967550034447367456878,
0.796197486967550034447367456878, 2.52442889579518216391449621291, 3.20585163256306441826577970590, 3.91854230155308754016351870095, 4.90129340528090247076056131033, 5.71872634371991770953386119123, 6.58246294038986299270139996042, 6.99426412218617473422211274605, 7.966391282386719564203088881500, 8.807539393288834856710357109022