L(s) = 1 | + 2.35·3-s + 5-s + 0.202·7-s + 2.54·9-s − 4.49·11-s − 0.979·13-s + 2.35·15-s + 0.563·17-s − 3.51·19-s + 0.477·21-s − 8.46·23-s + 25-s − 1.06·27-s − 3.31·29-s + 5.73·31-s − 10.5·33-s + 0.202·35-s − 8.46·37-s − 2.30·39-s − 6.59·41-s − 0.444·43-s + 2.54·45-s + 10.8·47-s − 6.95·49-s + 1.32·51-s − 6.63·53-s − 4.49·55-s + ⋯ |
L(s) = 1 | + 1.35·3-s + 0.447·5-s + 0.0766·7-s + 0.848·9-s − 1.35·11-s − 0.271·13-s + 0.608·15-s + 0.136·17-s − 0.806·19-s + 0.104·21-s − 1.76·23-s + 0.200·25-s − 0.205·27-s − 0.615·29-s + 1.02·31-s − 1.84·33-s + 0.0342·35-s − 1.39·37-s − 0.369·39-s − 1.03·41-s − 0.0677·43-s + 0.379·45-s + 1.58·47-s − 0.994·49-s + 0.185·51-s − 0.911·53-s − 0.605·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 7 | \( 1 - 0.202T + 7T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 + 0.979T + 13T^{2} \) |
| 17 | \( 1 - 0.563T + 17T^{2} \) |
| 19 | \( 1 + 3.51T + 19T^{2} \) |
| 23 | \( 1 + 8.46T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 + 0.444T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 - 0.633T + 61T^{2} \) |
| 67 | \( 1 - 4.51T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 0.361T + 73T^{2} \) |
| 79 | \( 1 - 2.63T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 6.68T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.961562400288021234623811863737, −7.25376270490065603437601703566, −6.32404204073243482935269584198, −5.55822224881183529122459783187, −4.76341958253666961918550738151, −3.87616085428272897808057328073, −3.08319142522861840056536955649, −2.31609462998700022563528253407, −1.80291184482599512488421391609, 0,
1.80291184482599512488421391609, 2.31609462998700022563528253407, 3.08319142522861840056536955649, 3.87616085428272897808057328073, 4.76341958253666961918550738151, 5.55822224881183529122459783187, 6.32404204073243482935269584198, 7.25376270490065603437601703566, 7.961562400288021234623811863737