L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s − 4·11-s + 13-s + 2·15-s + 17-s − 4·19-s + 4·21-s − 8·23-s − 25-s − 27-s − 2·29-s − 4·31-s + 4·33-s + 8·35-s − 6·37-s − 39-s − 6·41-s − 4·43-s − 2·45-s − 4·47-s + 9·49-s − 51-s + 14·53-s + 8·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.516·15-s + 0.242·17-s − 0.917·19-s + 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s + 1.35·35-s − 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s − 0.583·47-s + 9/7·49-s − 0.140·51-s + 1.92·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.42557877373924530600134195466, −6.76026394154398501303504624813, −5.98993941367257681574192896370, −5.48018620238465629134523253014, −4.38375315872604661607306220610, −3.72530619153079421336185659400, −3.02743634660319349609489976410, −1.88982522122326152562989294401, 0, 0,
1.88982522122326152562989294401, 3.02743634660319349609489976410, 3.72530619153079421336185659400, 4.38375315872604661607306220610, 5.48018620238465629134523253014, 5.98993941367257681574192896370, 6.76026394154398501303504624813, 7.42557877373924530600134195466