| L(s) = 1 | − 0.277·3-s − 5-s − 1.27·7-s − 2.92·9-s − 0.277·11-s + 3.92·13-s + 0.277·15-s + 5.47·17-s − 3.92·19-s + 0.354·21-s − 23-s + 25-s + 1.64·27-s − 0.799·29-s − 5.27·31-s + 0.0771·33-s + 1.27·35-s + 1.75·37-s − 1.08·39-s + 10.0·41-s − 4·43-s + 2.92·45-s + 8.95·47-s − 5.36·49-s − 1.52·51-s − 10.6·53-s + 0.277·55-s + ⋯ |
| L(s) = 1 | − 0.160·3-s − 0.447·5-s − 0.482·7-s − 0.974·9-s − 0.0837·11-s + 1.08·13-s + 0.0717·15-s + 1.32·17-s − 0.899·19-s + 0.0774·21-s − 0.208·23-s + 0.200·25-s + 0.316·27-s − 0.148·29-s − 0.947·31-s + 0.0134·33-s + 0.215·35-s + 0.288·37-s − 0.174·39-s + 1.56·41-s − 0.609·43-s + 0.435·45-s + 1.30·47-s − 0.766·49-s − 0.213·51-s − 1.46·53-s + 0.0374·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.277T + 3T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 11 | \( 1 + 0.277T + 11T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 + 3.92T + 19T^{2} \) |
| 29 | \( 1 + 0.799T + 29T^{2} \) |
| 31 | \( 1 + 5.27T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 8.95T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 8.08T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 9.32T + 71T^{2} \) |
| 73 | \( 1 - 4.55T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−7.76052754798791410546296301695, −6.76584354813495801637104681346, −6.04186274084694713922898891582, −5.67219781852308338634689644583, −4.71654444150467886719914668636, −3.74969646884856149229659182290, −3.30765875127388608263054605305, −2.35912164749504845072977544292, −1.12396867176856982253917638475, 0,
1.12396867176856982253917638475, 2.35912164749504845072977544292, 3.30765875127388608263054605305, 3.74969646884856149229659182290, 4.71654444150467886719914668636, 5.67219781852308338634689644583, 6.04186274084694713922898891582, 6.76584354813495801637104681346, 7.76052754798791410546296301695
