| L(s) = 1 | − 3·3-s + 2·5-s + 9·9-s − 12·11-s + 66·13-s − 6·15-s + 70·17-s − 92·19-s − 16·23-s − 121·25-s − 27·27-s − 122·29-s + 64·31-s + 36·33-s − 306·37-s − 198·39-s − 50·41-s − 20·43-s + 18·45-s − 176·47-s − 210·51-s + 526·53-s − 24·55-s + 276·57-s + 540·59-s + 818·61-s + 132·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.178·5-s + 1/3·9-s − 0.328·11-s + 1.40·13-s − 0.103·15-s + 0.998·17-s − 1.11·19-s − 0.145·23-s − 0.967·25-s − 0.192·27-s − 0.781·29-s + 0.370·31-s + 0.189·33-s − 1.35·37-s − 0.812·39-s − 0.190·41-s − 0.0709·43-s + 0.0596·45-s − 0.546·47-s − 0.576·51-s + 1.36·53-s − 0.0588·55-s + 0.641·57-s + 1.19·59-s + 1.71·61-s + 0.251·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 66 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 16 T + p^{3} T^{2} \) |
| 29 | \( 1 + 122 T + p^{3} T^{2} \) |
| 31 | \( 1 - 64 T + p^{3} T^{2} \) |
| 37 | \( 1 + 306 T + p^{3} T^{2} \) |
| 41 | \( 1 + 50 T + p^{3} T^{2} \) |
| 43 | \( 1 + 20 T + p^{3} T^{2} \) |
| 47 | \( 1 + 176 T + p^{3} T^{2} \) |
| 53 | \( 1 - 526 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 818 T + p^{3} T^{2} \) |
| 67 | \( 1 - 228 T + p^{3} T^{2} \) |
| 71 | \( 1 + 864 T + p^{3} T^{2} \) |
| 73 | \( 1 + 106 T + p^{3} T^{2} \) |
| 79 | \( 1 + 736 T + p^{3} T^{2} \) |
| 83 | \( 1 + 588 T + p^{3} T^{2} \) |
| 89 | \( 1 + 146 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1214 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.345549412487781754764313371575, −7.42847760699817966079261270042, −6.58173537921094919846176357572, −5.82874801899307607783467535035, −5.31600174251619490880476909369, −4.13023664485322659842060715098, −3.49059173603196032846325591814, −2.14732909749263846992258072088, −1.19004964692954393177442849285, 0,
1.19004964692954393177442849285, 2.14732909749263846992258072088, 3.49059173603196032846325591814, 4.13023664485322659842060715098, 5.31600174251619490880476909369, 5.82874801899307607783467535035, 6.58173537921094919846176357572, 7.42847760699817966079261270042, 8.345549412487781754764313371575
