| L(s) = 1 | + 3-s − 3·5-s − 3.46·7-s + 9-s + 1.73·13-s − 3·15-s − 1.73·17-s + 6.92·19-s − 3.46·21-s + 6·23-s + 4·25-s + 27-s + 1.73·29-s − 4·31-s + 10.3·35-s − 11·37-s + 1.73·39-s + 1.73·41-s + 3.46·43-s − 3·45-s + 4.99·49-s − 1.73·51-s − 9·53-s + 6.92·57-s + 6·59-s − 3.46·63-s − 5.19·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 1.30·7-s + 0.333·9-s + 0.480·13-s − 0.774·15-s − 0.420·17-s + 1.58·19-s − 0.755·21-s + 1.25·23-s + 0.800·25-s + 0.192·27-s + 0.321·29-s − 0.718·31-s + 1.75·35-s − 1.80·37-s + 0.277·39-s + 0.270·41-s + 0.528·43-s − 0.447·45-s + 0.714·49-s − 0.242·51-s − 1.23·53-s + 0.917·57-s + 0.781·59-s − 0.436·63-s − 0.644·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−7.71336859859283389226298923942, −7.08561334867391191814449146160, −6.64568740781964701391879423682, −5.55929857746052249522795208636, −4.72943165547924743838799258698, −3.65815787135977134704048895787, −3.46433167315444380293768872116, −2.66885812764643247826799173085, −1.18292495749787005783955020188, 0,
1.18292495749787005783955020188, 2.66885812764643247826799173085, 3.46433167315444380293768872116, 3.65815787135977134704048895787, 4.72943165547924743838799258698, 5.55929857746052249522795208636, 6.64568740781964701391879423682, 7.08561334867391191814449146160, 7.71336859859283389226298923942
