| L(s) = 1 | − 2-s + 0.509·3-s + 4-s + 0.844·5-s − 0.509·6-s + 7-s − 8-s − 2.74·9-s − 0.844·10-s + 4.76·11-s + 0.509·12-s + 1.82·13-s − 14-s + 0.430·15-s + 16-s + 2.72·17-s + 2.74·18-s − 5.56·19-s + 0.844·20-s + 0.509·21-s − 4.76·22-s + 6.28·23-s − 0.509·24-s − 4.28·25-s − 1.82·26-s − 2.92·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.294·3-s + 0.5·4-s + 0.377·5-s − 0.208·6-s + 0.377·7-s − 0.353·8-s − 0.913·9-s − 0.267·10-s + 1.43·11-s + 0.147·12-s + 0.505·13-s − 0.267·14-s + 0.111·15-s + 0.250·16-s + 0.660·17-s + 0.645·18-s − 1.27·19-s + 0.188·20-s + 0.111·21-s − 1.01·22-s + 1.31·23-s − 0.104·24-s − 0.857·25-s − 0.357·26-s − 0.563·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 0.509T + 3T^{2} \) |
| 5 | \( 1 - 0.844T + 5T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + 9.57T + 31T^{2} \) |
| 37 | \( 1 + 9.14T + 37T^{2} \) |
| 41 | \( 1 + 8.14T + 41T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 4.29T + 59T^{2} \) |
| 61 | \( 1 + 0.930T + 61T^{2} \) |
| 67 | \( 1 - 3.97T + 67T^{2} \) |
| 71 | \( 1 - 0.673T + 71T^{2} \) |
| 73 | \( 1 - 0.583T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 - 3.40T + 83T^{2} \) |
| 89 | \( 1 - 9.12T + 89T^{2} \) |
| 97 | \( 1 + 7.24T + 97T^{2} \) |
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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
−7.949993041449361862175735519861, −6.97839800276140401904839633262, −6.47148409075569844924647577218, −5.69892634117667832500295829493, −4.98441829760735507833325220721, −3.69796081196708671367953887943, −3.30946468352331492829064619617, −1.94890302108893317572669328316, −1.52802590210923875899833467645, 0,
1.52802590210923875899833467645, 1.94890302108893317572669328316, 3.30946468352331492829064619617, 3.69796081196708671367953887943, 4.98441829760735507833325220721, 5.69892634117667832500295829493, 6.47148409075569844924647577218, 6.97839800276140401904839633262, 7.949993041449361862175735519861
