| L(s) = 1 | + 1.90·2-s + 1.61·4-s − 1.61·5-s − 1.61·7-s − 0.726·8-s − 3.07·10-s + 0.236·11-s + 4.97·13-s − 3.07·14-s − 4.61·16-s − 0.236·17-s − 2.61·20-s + 0.449·22-s + 8.23·23-s − 2.38·25-s + 9.47·26-s − 2.61·28-s − 8.50·29-s − 7.33·31-s − 7.33·32-s − 0.449·34-s + 2.61·35-s − 5.25·37-s + 1.17·40-s + 3.52·41-s − 8.23·43-s + 0.381·44-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.809·4-s − 0.723·5-s − 0.611·7-s − 0.256·8-s − 0.973·10-s + 0.0711·11-s + 1.38·13-s − 0.822·14-s − 1.15·16-s − 0.0572·17-s − 0.585·20-s + 0.0957·22-s + 1.71·23-s − 0.476·25-s + 1.85·26-s − 0.494·28-s − 1.57·29-s − 1.31·31-s − 1.29·32-s − 0.0770·34-s + 0.442·35-s − 0.864·37-s + 0.185·40-s + 0.550·41-s − 1.25·43-s + 0.0575·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 - 4.97T + 13T^{2} \) |
| 17 | \( 1 + 0.236T + 17T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 + 7.33T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 8.23T + 43T^{2} \) |
| 47 | \( 1 + 8.38T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 + 8.95T + 67T^{2} \) |
| 71 | \( 1 + 8.78T + 71T^{2} \) |
| 73 | \( 1 - 0.708T + 73T^{2} \) |
| 79 | \( 1 - 0.171T + 79T^{2} \) |
| 83 | \( 1 - 9.94T + 83T^{2} \) |
| 89 | \( 1 + 0.898T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.253255441408486910385309226613, −7.23966136770815937426265441306, −6.63304730551083968877606358139, −5.85046023662720079549115292373, −5.18272585040832736454743164387, −4.27877964022742745734876731162, −3.43101414048794043506986659441, −3.26510144247503896915642680436, −1.71455338472824820316283796280, 0,
1.71455338472824820316283796280, 3.26510144247503896915642680436, 3.43101414048794043506986659441, 4.27877964022742745734876731162, 5.18272585040832736454743164387, 5.85046023662720079549115292373, 6.63304730551083968877606358139, 7.23966136770815937426265441306, 8.253255441408486910385309226613
