| L(s) = 1 | + 2·2-s + 3-s + 4·4-s + 2·6-s − 7·7-s + 8·8-s − 26·9-s − 35·11-s + 4·12-s − 58·13-s − 14·14-s + 16·16-s − 107·17-s − 52·18-s + 23·19-s − 7·21-s − 70·22-s + 200·23-s + 8·24-s − 116·26-s − 53·27-s − 28·28-s − 174·29-s + 76·31-s + 32·32-s − 35·33-s − 214·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.136·6-s − 0.377·7-s + 0.353·8-s − 0.962·9-s − 0.959·11-s + 0.0962·12-s − 1.23·13-s − 0.267·14-s + 1/4·16-s − 1.52·17-s − 0.680·18-s + 0.277·19-s − 0.0727·21-s − 0.678·22-s + 1.81·23-s + 0.0680·24-s − 0.874·26-s − 0.377·27-s − 0.188·28-s − 1.11·29-s + 0.440·31-s + 0.176·32-s − 0.184·33-s − 1.07·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 35 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 107 T + p^{3} T^{2} \) |
| 19 | \( 1 - 23 T + p^{3} T^{2} \) |
| 23 | \( 1 - 200 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 76 T + p^{3} T^{2} \) |
| 37 | \( 1 + 184 T + p^{3} T^{2} \) |
| 41 | \( 1 - 431 T + p^{3} T^{2} \) |
| 43 | \( 1 + 144 T + p^{3} T^{2} \) |
| 47 | \( 1 + 526 T + p^{3} T^{2} \) |
| 53 | \( 1 + 108 T + p^{3} T^{2} \) |
| 59 | \( 1 - 76 T + p^{3} T^{2} \) |
| 61 | \( 1 - 118 T + p^{3} T^{2} \) |
| 67 | \( 1 + 687 T + p^{3} T^{2} \) |
| 71 | \( 1 - 530 T + p^{3} T^{2} \) |
| 73 | \( 1 - 299 T + p^{3} T^{2} \) |
| 79 | \( 1 - 402 T + p^{3} T^{2} \) |
| 83 | \( 1 + 897 T + p^{3} T^{2} \) |
| 89 | \( 1 + 799 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1510 T + p^{3} T^{2} \) |
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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
−10.88471588442137775683138265827, −9.677541482273679394703243927881, −8.740836431583004797229753178833, −7.57719880710461324510626583428, −6.69571884999726671915396871732, −5.47512098103966863147142196146, −4.69212501962308766128563435529, −3.15084993041050397655266375817, −2.34931975864836021787385699129, 0,
2.34931975864836021787385699129, 3.15084993041050397655266375817, 4.69212501962308766128563435529, 5.47512098103966863147142196146, 6.69571884999726671915396871732, 7.57719880710461324510626583428, 8.740836431583004797229753178833, 9.677541482273679394703243927881, 10.88471588442137775683138265827
