| L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s − 5·11-s + 15-s − 17-s + 5·19-s + 3·21-s − 6·23-s + 25-s + 27-s − 3·29-s − 2·31-s − 5·33-s + 3·35-s + 7·37-s − 41-s − 4·43-s + 45-s − 3·47-s + 2·49-s − 51-s − 9·53-s − 5·55-s + 5·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.258·15-s − 0.242·17-s + 1.14·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s − 0.359·31-s − 0.870·33-s + 0.507·35-s + 1.15·37-s − 0.156·41-s − 0.609·43-s + 0.149·45-s − 0.437·47-s + 2/7·49-s − 0.140·51-s − 1.23·53-s − 0.674·55-s + 0.662·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.00375779781740, −12.33163286677771, −11.85996682888684, −11.42031212803149, −10.85812156110867, −10.62446935942727, −9.983948538345490, −9.509943527562430, −9.370455928554036, −8.418425900432187, −8.149275005235738, −7.923061673693911, −7.381783570848695, −6.925781671377630, −6.175452113219636, −5.711066294114000, −5.126731151182738, −4.967401094480347, −4.246688057796544, −3.750980084713729, −2.969298120368754, −2.683673471294621, −1.847811974159921, −1.767794691348873, −0.8478721209359522, 0,
0.8478721209359522, 1.767794691348873, 1.847811974159921, 2.683673471294621, 2.969298120368754, 3.750980084713729, 4.246688057796544, 4.967401094480347, 5.126731151182738, 5.711066294114000, 6.175452113219636, 6.925781671377630, 7.381783570848695, 7.923061673693911, 8.149275005235738, 8.418425900432187, 9.370455928554036, 9.509943527562430, 9.983948538345490, 10.62446935942727, 10.85812156110867, 11.42031212803149, 11.85996682888684, 12.33163286677771, 13.00375779781740
