| L(s) = 1 | − 3·3-s − 5·5-s − 32·7-s + 9·9-s + 36·11-s + 10·13-s + 15·15-s − 78·17-s + 140·19-s + 96·21-s + 192·23-s + 25·25-s − 27·27-s − 6·29-s + 16·31-s − 108·33-s + 160·35-s + 34·37-s − 30·39-s − 390·41-s − 52·43-s − 45·45-s − 408·47-s + 681·49-s + 234·51-s + 114·53-s − 180·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.72·7-s + 1/3·9-s + 0.986·11-s + 0.213·13-s + 0.258·15-s − 1.11·17-s + 1.69·19-s + 0.997·21-s + 1.74·23-s + 1/5·25-s − 0.192·27-s − 0.0384·29-s + 0.0926·31-s − 0.569·33-s + 0.772·35-s + 0.151·37-s − 0.123·39-s − 1.48·41-s − 0.184·43-s − 0.149·45-s − 1.26·47-s + 1.98·49-s + 0.642·51-s + 0.295·53-s − 0.441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 - 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 16 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 114 T + p^{3} T^{2} \) |
| 59 | \( 1 - 516 T + p^{3} T^{2} \) |
| 61 | \( 1 - 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 892 T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 646 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 + 732 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 - 2 p T + p^{3} T^{2} \) |
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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
−9.364713360330300328031263429551, −8.582222421768626830204004148675, −7.02286346889391393481707792677, −6.87885403228713974242085909329, −5.88937730575824317652420585112, −4.82491679264692457916136824758, −3.68493338026146904140285517198, −2.98737690724687525507730102827, −1.15186450444078892894826298361, 0,
1.15186450444078892894826298361, 2.98737690724687525507730102827, 3.68493338026146904140285517198, 4.82491679264692457916136824758, 5.88937730575824317652420585112, 6.87885403228713974242085909329, 7.02286346889391393481707792677, 8.582222421768626830204004148675, 9.364713360330300328031263429551
