| L(s) = 1 | + 1.98·3-s − 1.97·5-s + 3.06·7-s + 0.955·9-s − 3.47·11-s − 2.42·13-s − 3.92·15-s + 17-s + 2.06·19-s + 6.10·21-s + 3.27·23-s − 1.10·25-s − 4.06·27-s − 1.42·29-s + 1.29·31-s − 6.91·33-s − 6.05·35-s − 3.82·37-s − 4.82·39-s + 11.2·41-s − 3.17·43-s − 1.88·45-s − 5.65·47-s + 2.40·49-s + 1.98·51-s − 9.05·53-s + 6.86·55-s + ⋯ |
| L(s) = 1 | + 1.14·3-s − 0.882·5-s + 1.15·7-s + 0.318·9-s − 1.04·11-s − 0.672·13-s − 1.01·15-s + 0.242·17-s + 0.472·19-s + 1.33·21-s + 0.682·23-s − 0.220·25-s − 0.782·27-s − 0.264·29-s + 0.232·31-s − 1.20·33-s − 1.02·35-s − 0.629·37-s − 0.772·39-s + 1.75·41-s − 0.484·43-s − 0.281·45-s − 0.825·47-s + 0.343·49-s + 0.278·51-s − 1.24·53-s + 0.926·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 1.98T + 3T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 19 | \( 1 - 2.06T + 19T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 9.05T + 53T^{2} \) |
| 61 | \( 1 + 7.52T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 - 7.96T + 79T^{2} \) |
| 83 | \( 1 - 3.39T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.61924174396169134085345784891, −7.34811312349531969319477359141, −6.06563313615334301532317690801, −5.10908465787095731155450470397, −4.70140682461328760123471499171, −3.77830298330053254816519065767, −3.04324465856033283716917175006, −2.40061207156069619731441434090, −1.45309837128122083046052095910, 0,
1.45309837128122083046052095910, 2.40061207156069619731441434090, 3.04324465856033283716917175006, 3.77830298330053254816519065767, 4.70140682461328760123471499171, 5.10908465787095731155450470397, 6.06563313615334301532317690801, 7.34811312349531969319477359141, 7.61924174396169134085345784891
