| L(s) = 1 | − 3-s − 5-s − 3.12·7-s + 9-s − 2·11-s + 3.12·13-s + 15-s + 7.12·17-s + 3.12·19-s + 3.12·21-s + 3.12·23-s + 25-s − 27-s − 8.24·29-s + 1.12·31-s + 2·33-s + 3.12·35-s − 3.12·37-s − 3.12·39-s − 2·41-s − 10.2·43-s − 45-s − 4.87·47-s + 2.75·49-s − 7.12·51-s − 10·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.18·7-s + 0.333·9-s − 0.603·11-s + 0.866·13-s + 0.258·15-s + 1.72·17-s + 0.716·19-s + 0.681·21-s + 0.651·23-s + 0.200·25-s − 0.192·27-s − 1.53·29-s + 0.201·31-s + 0.348·33-s + 0.527·35-s − 0.513·37-s − 0.500·39-s − 0.312·41-s − 1.56·43-s − 0.149·45-s − 0.711·47-s + 0.393·49-s − 0.997·51-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 4.87T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 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
−8.870590244751674595569211086365, −7.87162060268002339101369770475, −7.26041274963807617144640624302, −6.34911841690196623403282083067, −5.65243634256258270361168155659, −4.88267671041587512861156794536, −3.49225941958914894474644998999, −3.21543304264118352878127473748, −1.38280231511189094058156625798, 0,
1.38280231511189094058156625798, 3.21543304264118352878127473748, 3.49225941958914894474644998999, 4.88267671041587512861156794536, 5.65243634256258270361168155659, 6.34911841690196623403282083067, 7.26041274963807617144640624302, 7.87162060268002339101369770475, 8.870590244751674595569211086365
