| L(s) = 1 | − 2.26·2-s − 1.07·3-s + 3.15·4-s + 5-s + 2.44·6-s + 0.653·7-s − 2.61·8-s − 1.84·9-s − 2.26·10-s + 5.79·11-s − 3.39·12-s − 3.07·13-s − 1.48·14-s − 1.07·15-s − 0.371·16-s + 0.143·17-s + 4.18·18-s + 0.822·19-s + 3.15·20-s − 0.702·21-s − 13.1·22-s − 6.79·23-s + 2.81·24-s + 25-s + 6.98·26-s + 5.21·27-s + 2.05·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 0.621·3-s + 1.57·4-s + 0.447·5-s + 0.996·6-s + 0.246·7-s − 0.923·8-s − 0.614·9-s − 0.717·10-s + 1.74·11-s − 0.978·12-s − 0.853·13-s − 0.396·14-s − 0.277·15-s − 0.0929·16-s + 0.0347·17-s + 0.985·18-s + 0.188·19-s + 0.704·20-s − 0.153·21-s − 2.80·22-s − 1.41·23-s + 0.573·24-s + 0.200·25-s + 1.36·26-s + 1.00·27-s + 0.389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5620124078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5620124078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 + 1.07T + 3T^{2} \) |
| 7 | \( 1 - 0.653T + 7T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 17 | \( 1 - 0.143T + 17T^{2} \) |
| 19 | \( 1 - 0.822T + 19T^{2} \) |
| 23 | \( 1 + 6.79T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 - 7.38T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 5.84T + 43T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 - 9.60T + 59T^{2} \) |
| 61 | \( 1 + 0.815T + 61T^{2} \) |
| 67 | \( 1 - 2.63T + 67T^{2} \) |
| 71 | \( 1 - 0.948T + 71T^{2} \) |
| 73 | \( 1 - 8.40T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 89 | \( 1 + 1.19T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−11.10220827564567782442961107531, −10.10132656660113954916654991395, −9.427114182133227645145490070543, −8.674483021736079639247117088737, −7.69246639747450893259974858596, −6.62978751337234777717602326275, −5.90869102968939921746560918106, −4.39532636557707521904972703132, −2.41056375537551807350770991689, −0.961001017275539865942656682467,
0.961001017275539865942656682467, 2.41056375537551807350770991689, 4.39532636557707521904972703132, 5.90869102968939921746560918106, 6.62978751337234777717602326275, 7.69246639747450893259974858596, 8.674483021736079639247117088737, 9.427114182133227645145490070543, 10.10132656660113954916654991395, 11.10220827564567782442961107531
