| L(s) = 1 | − 6·5-s + 7·7-s − 36·11-s + 62·13-s − 114·17-s − 76·19-s + 24·23-s − 89·25-s − 54·29-s − 112·31-s − 42·35-s − 178·37-s − 378·41-s − 172·43-s + 192·47-s + 49·49-s + 402·53-s + 216·55-s − 396·59-s + 254·61-s − 372·65-s − 1.01e3·67-s − 840·71-s + 890·73-s − 252·77-s + 80·79-s + 108·83-s + ⋯ |
| L(s) = 1 | − 0.536·5-s + 0.377·7-s − 0.986·11-s + 1.32·13-s − 1.62·17-s − 0.917·19-s + 0.217·23-s − 0.711·25-s − 0.345·29-s − 0.648·31-s − 0.202·35-s − 0.790·37-s − 1.43·41-s − 0.609·43-s + 0.595·47-s + 1/7·49-s + 1.04·53-s + 0.529·55-s − 0.873·59-s + 0.533·61-s − 0.709·65-s − 1.84·67-s − 1.40·71-s + 1.42·73-s − 0.372·77-s + 0.113·79-s + 0.142·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 24 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 192 T + p^{3} T^{2} \) |
| 53 | \( 1 - 402 T + p^{3} T^{2} \) |
| 59 | \( 1 + 396 T + p^{3} T^{2} \) |
| 61 | \( 1 - 254 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1012 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 890 T + p^{3} T^{2} \) |
| 79 | \( 1 - 80 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1638 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1010 T + p^{3} T^{2} \) |
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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
−11.02851742986436830150646304293, −10.48109944032639874377566182827, −8.909415316301294582046174060833, −8.306413665781800601275156479431, −7.18515319544684163584848087811, −6.03509074832949923859509344246, −4.74695254805227976499825147575, −3.63460833012397145434805269090, −1.99987871287688285887259941607, 0,
1.99987871287688285887259941607, 3.63460833012397145434805269090, 4.74695254805227976499825147575, 6.03509074832949923859509344246, 7.18515319544684163584848087811, 8.306413665781800601275156479431, 8.909415316301294582046174060833, 10.48109944032639874377566182827, 11.02851742986436830150646304293
