| 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
| \(L(1)\) |
\(\approx\) |
\(1.226620115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226620115\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 8T + 97T^{2} \) |
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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
−11.00418174970039750289559385072, −10.04203083922318676013244980209, −9.171106386262901820522468309528, −8.504509402954959248433237085570, −6.61387298753804977066819083208, −6.39786281520133230964526164360, −5.59569989859156343397361984183, −4.17878088743430670818200841248, −2.93642251168937868637046890158, −1.12736751813594213332646162785,
1.12736751813594213332646162785, 2.93642251168937868637046890158, 4.17878088743430670818200841248, 5.59569989859156343397361984183, 6.39786281520133230964526164360, 6.61387298753804977066819083208, 8.504509402954959248433237085570, 9.171106386262901820522468309528, 10.04203083922318676013244980209, 11.00418174970039750289559385072
