| L(s) = 1 | + 5-s − 1.41·7-s + 1.41·11-s + 2.24·13-s − 5.24·17-s − 1.24·19-s + 8.65·23-s + 25-s − 4.24·29-s + 4.07·31-s − 1.41·35-s − 6.48·37-s + 6.24·41-s + 10.2·43-s + 0.343·47-s − 5·49-s + 1.24·53-s + 1.41·55-s − 1.07·59-s + 61-s + 2.24·65-s + 0.343·67-s + 10.5·71-s + 4.24·73-s − 2.00·77-s − 1.58·79-s + 6.17·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.534·7-s + 0.426·11-s + 0.621·13-s − 1.27·17-s − 0.285·19-s + 1.80·23-s + 0.200·25-s − 0.787·29-s + 0.731·31-s − 0.239·35-s − 1.06·37-s + 0.974·41-s + 1.56·43-s + 0.0500·47-s − 0.714·49-s + 0.170·53-s + 0.190·55-s − 0.139·59-s + 0.128·61-s + 0.278·65-s + 0.0419·67-s + 1.25·71-s + 0.496·73-s − 0.227·77-s − 0.178·79-s + 0.677·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.991770105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991770105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 8.65T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 4.07T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 1.24T + 53T^{2} \) |
| 59 | \( 1 + 1.07T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 0.343T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 - 2.24T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.603433447795772215469143452813, −7.56142541047912341153826129082, −6.73377233403310342526347958084, −6.35337694053796298810875010341, −5.47359318218913846475187518518, −4.62503816577945621161583703515, −3.79275608903210650264588527978, −2.90306032285899687091480646873, −1.98439807781600925781737027473, −0.802917093643281229444059871538,
0.802917093643281229444059871538, 1.98439807781600925781737027473, 2.90306032285899687091480646873, 3.79275608903210650264588527978, 4.62503816577945621161583703515, 5.47359318218913846475187518518, 6.35337694053796298810875010341, 6.73377233403310342526347958084, 7.56142541047912341153826129082, 8.603433447795772215469143452813
