| L(s) = 1 | + 3-s + 5-s + 5.12·7-s + 9-s + 2·11-s + 5.12·13-s + 15-s − 1.12·17-s + 5.12·19-s + 5.12·21-s − 5.12·23-s + 25-s + 27-s − 8.24·29-s − 7.12·31-s + 2·33-s + 5.12·35-s − 5.12·37-s + 5.12·39-s − 2·41-s − 6.24·43-s + 45-s − 13.1·47-s + 19.2·49-s − 1.12·51-s + 10·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.93·7-s + 0.333·9-s + 0.603·11-s + 1.42·13-s + 0.258·15-s − 0.272·17-s + 1.17·19-s + 1.11·21-s − 1.06·23-s + 0.200·25-s + 0.192·27-s − 1.53·29-s − 1.27·31-s + 0.348·33-s + 0.865·35-s − 0.842·37-s + 0.820·39-s − 0.312·41-s − 0.952·43-s + 0.149·45-s − 1.91·47-s + 2.74·49-s − 0.157·51-s + 1.37·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.195090563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.195090563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.938212050614217561674344687628, −8.536944800966829093560961497328, −7.75203957098456649763370360724, −7.03588215097266470636524751115, −5.84583346946819355992170483941, −5.21603207312641333367231469499, −4.17361941381657364250233413630, −3.43103357452412713358626233328, −1.85249753814038468668508443602, −1.49221599261218438514184861275,
1.49221599261218438514184861275, 1.85249753814038468668508443602, 3.43103357452412713358626233328, 4.17361941381657364250233413630, 5.21603207312641333367231469499, 5.84583346946819355992170483941, 7.03588215097266470636524751115, 7.75203957098456649763370360724, 8.536944800966829093560961497328, 8.938212050614217561674344687628
