| L(s) = 1 | − 3-s − 5-s + 0.512·7-s + 9-s − 1.76·11-s − 4.95·13-s + 15-s + 7.97·17-s + 1.93·19-s − 0.512·21-s + 23-s + 25-s − 27-s − 9.16·29-s + 4.20·31-s + 1.76·33-s − 0.512·35-s + 6.37·37-s + 4.95·39-s + 4.27·41-s − 12.8·43-s − 45-s − 1.93·47-s − 6.73·49-s − 7.97·51-s + 4.60·53-s + 1.76·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.193·7-s + 0.333·9-s − 0.532·11-s − 1.37·13-s + 0.258·15-s + 1.93·17-s + 0.442·19-s − 0.111·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s − 1.70·29-s + 0.755·31-s + 0.307·33-s − 0.0866·35-s + 1.04·37-s + 0.793·39-s + 0.668·41-s − 1.96·43-s − 0.149·45-s − 0.281·47-s − 0.962·49-s − 1.11·51-s + 0.632·53-s + 0.237·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 0.512T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 17 | \( 1 - 7.97T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 29 | \( 1 + 9.16T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 - 6.37T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 1.93T + 47T^{2} \) |
| 53 | \( 1 - 4.60T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 8.65T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 2.27T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 + 6.91T + 83T^{2} \) |
| 89 | \( 1 + 1.69T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.155403703596304798138380386565, −7.64775199125265453399953741694, −7.11645083037279646356689229399, −5.99110806161098016085784450967, −5.26474297359390919074306434316, −4.71036397242113196717869872928, −3.59222980042004935753254514904, −2.70799578435897261569244533211, −1.35530742836390379693247455715, 0,
1.35530742836390379693247455715, 2.70799578435897261569244533211, 3.59222980042004935753254514904, 4.71036397242113196717869872928, 5.26474297359390919074306434316, 5.99110806161098016085784450967, 7.11645083037279646356689229399, 7.64775199125265453399953741694, 8.155403703596304798138380386565
