L(s) = 1 | − 2-s − 1.60·3-s + 4-s + 1.84·5-s + 1.60·6-s − 2.27·7-s − 8-s − 0.408·9-s − 1.84·10-s + 0.886·11-s − 1.60·12-s + 2.28·13-s + 2.27·14-s − 2.97·15-s + 16-s − 0.225·17-s + 0.408·18-s + 2.70·19-s + 1.84·20-s + 3.65·21-s − 0.886·22-s − 3.08·23-s + 1.60·24-s − 1.58·25-s − 2.28·26-s + 5.48·27-s − 2.27·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.929·3-s + 0.5·4-s + 0.826·5-s + 0.657·6-s − 0.858·7-s − 0.353·8-s − 0.136·9-s − 0.584·10-s + 0.267·11-s − 0.464·12-s + 0.632·13-s + 0.606·14-s − 0.768·15-s + 0.250·16-s − 0.0545·17-s + 0.0962·18-s + 0.619·19-s + 0.413·20-s + 0.797·21-s − 0.188·22-s − 0.644·23-s + 0.328·24-s − 0.316·25-s − 0.447·26-s + 1.05·27-s − 0.429·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
good | 3 | \( 1 + 1.60T + 3T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 - 0.886T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 + 0.225T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 37 | \( 1 + 9.31T + 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 - 8.74T + 53T^{2} \) |
| 59 | \( 1 - 1.10T + 59T^{2} \) |
| 61 | \( 1 + 4.06T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 + 8.92T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 0.473T + 89T^{2} \) |
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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
−7.62897580768607879399055761574, −6.91462565337324186328755091773, −6.15466537544504106668294546735, −5.88835841463245590684688245025, −5.18388425708523292013022408914, −3.97107343731063815122033642050, −3.10268175660847671994391682095, −2.12515114987833900473808690558, −1.10149265071351921179369535860, 0,
1.10149265071351921179369535860, 2.12515114987833900473808690558, 3.10268175660847671994391682095, 3.97107343731063815122033642050, 5.18388425708523292013022408914, 5.88835841463245590684688245025, 6.15466537544504106668294546735, 6.91462565337324186328755091773, 7.62897580768607879399055761574