| L(s) = 1 | − 0.460·3-s + 0.528·5-s − 1.88·7-s − 2.78·9-s + 4.52·11-s − 4.67·13-s − 0.243·15-s + 7.22·17-s − 2.05·19-s + 0.868·21-s − 0.329·23-s − 4.72·25-s + 2.66·27-s − 3.05·29-s − 7.53·31-s − 2.08·33-s − 0.996·35-s − 8.39·37-s + 2.15·39-s − 5.74·41-s + 3.97·43-s − 1.47·45-s − 3.13·47-s − 3.44·49-s − 3.32·51-s − 4.78·53-s + 2.38·55-s + ⋯ |
| L(s) = 1 | − 0.265·3-s + 0.236·5-s − 0.712·7-s − 0.929·9-s + 1.36·11-s − 1.29·13-s − 0.0628·15-s + 1.75·17-s − 0.470·19-s + 0.189·21-s − 0.0687·23-s − 0.944·25-s + 0.513·27-s − 0.567·29-s − 1.35·31-s − 0.362·33-s − 0.168·35-s − 1.37·37-s + 0.344·39-s − 0.897·41-s + 0.606·43-s − 0.219·45-s − 0.456·47-s − 0.491·49-s − 0.466·51-s − 0.656·53-s + 0.322·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 0.460T + 3T^{2} \) |
| 5 | \( 1 - 0.528T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 + 0.329T + 23T^{2} \) |
| 29 | \( 1 + 3.05T + 29T^{2} \) |
| 31 | \( 1 + 7.53T + 31T^{2} \) |
| 37 | \( 1 + 8.39T + 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 - 3.97T + 43T^{2} \) |
| 47 | \( 1 + 3.13T + 47T^{2} \) |
| 53 | \( 1 + 4.78T + 53T^{2} \) |
| 59 | \( 1 - 8.57T + 59T^{2} \) |
| 61 | \( 1 + 4.95T + 61T^{2} \) |
| 67 | \( 1 + 4.60T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.43T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 4.86T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 1.79T + 97T^{2} \) |
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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
−9.441699695528157990447593702073, −8.489339091220696947055284921741, −7.48523969389589892371847288724, −6.69928366505691856288510222749, −5.82890541786638879745261538703, −5.22073686574336791315416641421, −3.86601576930200172431683344133, −3.08471518501592427061187269296, −1.73877678076197676665562785960, 0,
1.73877678076197676665562785960, 3.08471518501592427061187269296, 3.86601576930200172431683344133, 5.22073686574336791315416641421, 5.82890541786638879745261538703, 6.69928366505691856288510222749, 7.48523969389589892371847288724, 8.489339091220696947055284921741, 9.441699695528157990447593702073
