| L(s) = 1 | + 2.37·2-s − 2.28·3-s + 3.62·4-s − 5.41·6-s − 1.63·7-s + 3.84·8-s + 2.22·9-s + 2.72·11-s − 8.27·12-s − 6.19·13-s − 3.88·14-s + 1.87·16-s + 3.12·17-s + 5.26·18-s + 3.74·21-s + 6.46·22-s − 7.29·23-s − 8.78·24-s − 14.6·26-s + 1.77·27-s − 5.92·28-s + 2.22·29-s + 4.42·31-s − 3.25·32-s − 6.23·33-s + 7.40·34-s + 8.04·36-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 1.31·3-s + 1.81·4-s − 2.21·6-s − 0.618·7-s + 1.35·8-s + 0.740·9-s + 0.822·11-s − 2.38·12-s − 1.71·13-s − 1.03·14-s + 0.468·16-s + 0.757·17-s + 1.24·18-s + 0.816·21-s + 1.37·22-s − 1.52·23-s − 1.79·24-s − 2.88·26-s + 0.342·27-s − 1.12·28-s + 0.413·29-s + 0.795·31-s − 0.574·32-s − 1.08·33-s + 1.27·34-s + 1.34·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.623599545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.623599545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 3 | \( 1 + 2.28T + 3T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 + 2.30T + 47T^{2} \) |
| 53 | \( 1 - 6.36T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 - 0.670T + 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 - 4.12T + 79T^{2} \) |
| 83 | \( 1 + 6.42T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 0.129T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.26265505215825784457371679224, −6.63968763092900338629561307391, −6.18796300892144660029675041236, −5.62309290317772359218676955379, −4.97893416705272650885382796717, −4.43428047050446766120726029420, −3.71624672457802535564499044427, −2.85579344832419685745046378243, −2.03665518519476325296488351851, −0.62067047550699641610169742652,
0.62067047550699641610169742652, 2.03665518519476325296488351851, 2.85579344832419685745046378243, 3.71624672457802535564499044427, 4.43428047050446766120726029420, 4.97893416705272650885382796717, 5.62309290317772359218676955379, 6.18796300892144660029675041236, 6.63968763092900338629561307391, 7.26265505215825784457371679224
