| L(s) = 1 | + 0.175·2-s − 0.522·3-s − 1.96·4-s − 0.0832·5-s − 0.0917·6-s + 1.97·7-s − 0.697·8-s − 2.72·9-s − 0.0146·10-s + 4.76·11-s + 1.02·12-s − 2.67·13-s + 0.346·14-s + 0.0435·15-s + 3.81·16-s − 7.24·17-s − 0.478·18-s − 6.71·19-s + 0.163·20-s − 1.03·21-s + 0.837·22-s + 2.79·23-s + 0.364·24-s − 4.99·25-s − 0.469·26-s + 2.99·27-s − 3.88·28-s + ⋯ |
| L(s) = 1 | + 0.124·2-s − 0.301·3-s − 0.984·4-s − 0.0372·5-s − 0.0374·6-s + 0.745·7-s − 0.246·8-s − 0.908·9-s − 0.00462·10-s + 1.43·11-s + 0.297·12-s − 0.742·13-s + 0.0925·14-s + 0.0112·15-s + 0.953·16-s − 1.75·17-s − 0.112·18-s − 1.54·19-s + 0.0366·20-s − 0.225·21-s + 0.178·22-s + 0.582·23-s + 0.0743·24-s − 0.998·25-s − 0.0921·26-s + 0.576·27-s − 0.734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7822459584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7822459584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 - 0.175T + 2T^{2} \) |
| 3 | \( 1 + 0.522T + 3T^{2} \) |
| 5 | \( 1 + 0.0832T + 5T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 7.24T + 17T^{2} \) |
| 19 | \( 1 + 6.71T + 19T^{2} \) |
| 23 | \( 1 - 2.79T + 23T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 + 0.437T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 + 1.98T + 43T^{2} \) |
| 47 | \( 1 - 1.77T + 47T^{2} \) |
| 53 | \( 1 + 0.0458T + 53T^{2} \) |
| 59 | \( 1 - 8.22T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 3.27T + 67T^{2} \) |
| 71 | \( 1 + 2.77T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 - 1.21T + 83T^{2} \) |
| 89 | \( 1 + 3.27T + 89T^{2} \) |
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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
−7.908674877499836932103699896991, −6.81996365200405685928990890332, −6.35857083484611289595925632848, −5.52824212073546151025876067510, −4.90246560017856872022711986350, −4.18361768396799672388747968160, −3.82366193524616824698696556466, −2.52015232204871520075233670414, −1.73882880048964195937166019031, −0.41767152421237360295518033769,
0.41767152421237360295518033769, 1.73882880048964195937166019031, 2.52015232204871520075233670414, 3.82366193524616824698696556466, 4.18361768396799672388747968160, 4.90246560017856872022711986350, 5.52824212073546151025876067510, 6.35857083484611289595925632848, 6.81996365200405685928990890332, 7.908674877499836932103699896991
