| L(s) = 1 | − 0.745·3-s − 5-s − 4.42·7-s − 2.44·9-s + 5.87·11-s − 3.25·13-s + 0.745·15-s + 2.93·17-s + 2.50·19-s + 3.30·21-s + 0.745·23-s + 25-s + 4.06·27-s + 29-s − 4.29·31-s − 4.37·33-s + 4.42·35-s + 6.85·37-s + 2.42·39-s + 7.87·41-s + 5.44·43-s + 2.44·45-s − 10.3·47-s + 12.6·49-s − 2.18·51-s − 0.0481·53-s − 5.87·55-s + ⋯ |
| L(s) = 1 | − 0.430·3-s − 0.447·5-s − 1.67·7-s − 0.814·9-s + 1.77·11-s − 0.902·13-s + 0.192·15-s + 0.711·17-s + 0.575·19-s + 0.720·21-s + 0.155·23-s + 0.200·25-s + 0.781·27-s + 0.185·29-s − 0.771·31-s − 0.762·33-s + 0.748·35-s + 1.12·37-s + 0.388·39-s + 1.22·41-s + 0.830·43-s + 0.364·45-s − 1.51·47-s + 1.80·49-s − 0.306·51-s − 0.00661·53-s − 0.791·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.745T + 3T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 5.87T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 23 | \( 1 - 0.745T + 23T^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 - 7.87T + 41T^{2} \) |
| 43 | \( 1 - 5.44T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 0.0481T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 1.60T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 7.74T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 - 3.97T + 79T^{2} \) |
| 83 | \( 1 - 3.87T + 83T^{2} \) |
| 89 | \( 1 - 0.766T + 89T^{2} \) |
| 97 | \( 1 + 0.270T + 97T^{2} \) |
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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
−7.77199211333317055699301470465, −7.17378229782199855272069658246, −6.27864207814005303959715644736, −6.08643201310947882462276346309, −5.03069152110892335687402256213, −4.05325200227098068848444224114, −3.34264116044636254376184388872, −2.69216002068966009870271130844, −1.09182934672051063617201799465, 0,
1.09182934672051063617201799465, 2.69216002068966009870271130844, 3.34264116044636254376184388872, 4.05325200227098068848444224114, 5.03069152110892335687402256213, 6.08643201310947882462276346309, 6.27864207814005303959715644736, 7.17378229782199855272069658246, 7.77199211333317055699301470465
