| L(s) = 1 | − 1.41·3-s − 2·5-s + 2.82·7-s − 0.999·9-s + 4.24·11-s − 6·13-s + 2.82·15-s − 4.24·19-s − 4.00·21-s − 8.48·23-s − 25-s + 5.65·27-s − 2·29-s − 5.65·31-s − 6·33-s − 5.65·35-s − 6·37-s + 8.48·39-s + 6·41-s − 4.24·43-s + 1.99·45-s + 1.00·49-s + 2·53-s − 8.48·55-s + 6·57-s + 1.41·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.894·5-s + 1.06·7-s − 0.333·9-s + 1.27·11-s − 1.66·13-s + 0.730·15-s − 0.973·19-s − 0.872·21-s − 1.76·23-s − 0.200·25-s + 1.08·27-s − 0.371·29-s − 1.01·31-s − 1.04·33-s − 0.956·35-s − 0.986·37-s + 1.35·39-s + 0.937·41-s − 0.646·43-s + 0.298·45-s + 0.142·49-s + 0.274·53-s − 1.14·55-s + 0.794·57-s + 0.184·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
| 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
−10.72777274474817207367752876802, −9.635349597153063340889306226847, −8.518825838212268140126510294240, −7.74175324581225334869780225691, −6.80409189002055853973715740956, −5.70181878856779292233128223900, −4.69684119897158394604115836867, −3.87700045654179342788404540255, −2.00670986314658715682425526222, 0,
2.00670986314658715682425526222, 3.87700045654179342788404540255, 4.69684119897158394604115836867, 5.70181878856779292233128223900, 6.80409189002055853973715740956, 7.74175324581225334869780225691, 8.518825838212268140126510294240, 9.635349597153063340889306226847, 10.72777274474817207367752876802
