| L(s) = 1 | − 1.53·2-s + 0.347·4-s + 3.53·5-s + 0.347·7-s + 2.53·8-s − 5.41·10-s − 1.75·11-s − 3.29·13-s − 0.532·14-s − 4.57·16-s + 1.53·19-s + 1.22·20-s + 2.69·22-s + 2.81·23-s + 7.47·25-s + 5.04·26-s + 0.120·28-s − 1.18·29-s − 7.10·31-s + 1.94·32-s + 1.22·35-s + 3.92·37-s − 2.34·38-s + 8.94·40-s + 4.92·41-s + 10.9·43-s − 0.610·44-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.173·4-s + 1.57·5-s + 0.131·7-s + 0.895·8-s − 1.71·10-s − 0.530·11-s − 0.912·13-s − 0.142·14-s − 1.14·16-s + 0.351·19-s + 0.274·20-s + 0.574·22-s + 0.587·23-s + 1.49·25-s + 0.988·26-s + 0.0227·28-s − 0.220·29-s − 1.27·31-s + 0.343·32-s + 0.207·35-s + 0.644·37-s − 0.380·38-s + 1.41·40-s + 0.768·41-s + 1.67·43-s − 0.0920·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223017314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223017314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 - 0.347T + 7T^{2} \) |
| 11 | \( 1 + 1.75T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 - 8.36T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 67 | \( 1 - 8.07T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 1.70T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 + 6.41T + 89T^{2} \) |
| 97 | \( 1 + 4.08T + 97T^{2} \) |
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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
−9.162566669514389227814185956835, −8.265490198200679693894224366354, −7.42360661984216685163376243015, −6.85372768148540271430247286217, −5.63519372814090588965817849592, −5.26009453503004804062573838516, −4.19204571858054304736805650691, −2.63859168933236895116316281585, −1.98322058079706519795374553487, −0.836847174517451435522936574431,
0.836847174517451435522936574431, 1.98322058079706519795374553487, 2.63859168933236895116316281585, 4.19204571858054304736805650691, 5.26009453503004804062573838516, 5.63519372814090588965817849592, 6.85372768148540271430247286217, 7.42360661984216685163376243015, 8.265490198200679693894224366354, 9.162566669514389227814185956835
