| L(s) = 1 | + 2·3-s + 14·5-s − 23·9-s − 20·11-s − 6·13-s + 28·15-s + 20·17-s + 102·19-s − 124·23-s + 71·25-s − 100·27-s − 78·29-s + 236·31-s − 40·33-s + 66·37-s − 12·39-s + 268·41-s + 132·43-s − 322·45-s + 516·47-s + 40·51-s − 354·53-s − 280·55-s + 204·57-s + 438·59-s + 486·61-s − 84·65-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.25·5-s − 0.851·9-s − 0.548·11-s − 0.128·13-s + 0.481·15-s + 0.285·17-s + 1.23·19-s − 1.12·23-s + 0.567·25-s − 0.712·27-s − 0.499·29-s + 1.36·31-s − 0.211·33-s + 0.293·37-s − 0.0492·39-s + 1.02·41-s + 0.468·43-s − 1.06·45-s + 1.60·47-s + 0.109·51-s − 0.917·53-s − 0.686·55-s + 0.474·57-s + 0.966·59-s + 1.02·61-s − 0.160·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.959996816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.959996816\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 20 T + p^{3} T^{2} \) |
| 19 | \( 1 - 102 T + p^{3} T^{2} \) |
| 23 | \( 1 + 124 T + p^{3} T^{2} \) |
| 29 | \( 1 + 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 66 T + p^{3} T^{2} \) |
| 41 | \( 1 - 268 T + p^{3} T^{2} \) |
| 43 | \( 1 - 132 T + p^{3} T^{2} \) |
| 47 | \( 1 - 516 T + p^{3} T^{2} \) |
| 53 | \( 1 + 354 T + p^{3} T^{2} \) |
| 59 | \( 1 - 438 T + p^{3} T^{2} \) |
| 61 | \( 1 - 486 T + p^{3} T^{2} \) |
| 67 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 248 T + p^{3} T^{2} \) |
| 73 | \( 1 - 768 T + p^{3} T^{2} \) |
| 79 | \( 1 - 192 T + p^{3} T^{2} \) |
| 83 | \( 1 - 294 T + p^{3} T^{2} \) |
| 89 | \( 1 - 80 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1404 T + p^{3} T^{2} \) |
| 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
−9.279987511990875582287263063412, −8.220068873244135222967229224525, −7.66542545213885348270376701325, −6.48778241377503888021997484004, −5.69463702879367596189944056212, −5.23362621806134872238074237486, −3.87687539180523621674586939868, −2.74022099412189180957866670337, −2.16776182887618265605523738707, −0.811260449215143362668711563706,
0.811260449215143362668711563706, 2.16776182887618265605523738707, 2.74022099412189180957866670337, 3.87687539180523621674586939868, 5.23362621806134872238074237486, 5.69463702879367596189944056212, 6.48778241377503888021997484004, 7.66542545213885348270376701325, 8.220068873244135222967229224525, 9.279987511990875582287263063412
