| L(s) = 1 | + 1.19·3-s + 5-s − 1.83·7-s − 1.57·9-s − 1.25·13-s + 1.19·15-s + 6.44·17-s + 1.25·19-s − 2.19·21-s − 4.35·23-s + 25-s − 5.46·27-s − 0.726·29-s − 11.0·31-s − 1.83·35-s − 11.5·37-s − 1.49·39-s + 7.92·41-s + 0.606·43-s − 1.57·45-s − 3.98·47-s − 3.62·49-s + 7.69·51-s + 6.83·53-s + 1.49·57-s − 1.41·59-s − 9.75·61-s + ⋯ |
| L(s) = 1 | + 0.689·3-s + 0.447·5-s − 0.694·7-s − 0.525·9-s − 0.348·13-s + 0.308·15-s + 1.56·17-s + 0.287·19-s − 0.478·21-s − 0.908·23-s + 0.200·25-s − 1.05·27-s − 0.134·29-s − 1.98·31-s − 0.310·35-s − 1.89·37-s − 0.239·39-s + 1.23·41-s + 0.0925·43-s − 0.234·45-s − 0.580·47-s − 0.517·49-s + 1.07·51-s + 0.938·53-s + 0.197·57-s − 0.183·59-s − 1.24·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 + 0.726T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 7.92T + 41T^{2} \) |
| 43 | \( 1 - 0.606T + 43T^{2} \) |
| 47 | \( 1 + 3.98T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 9.75T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 6.51T + 79T^{2} \) |
| 83 | \( 1 - 0.644T + 83T^{2} \) |
| 89 | \( 1 - 9.39T + 89T^{2} \) |
| 97 | \( 1 + 4.66T + 97T^{2} \) |
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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
−7.80564621191749223666525872411, −7.42070894767394719178986604652, −6.41777743316307942878589698341, −5.66146745069945851023919856596, −5.19936486244070317479880685098, −3.80527778884990097102940261143, −3.34429622718569908729876339985, −2.49561686725426978994739439666, −1.56482899299555443109336624908, 0,
1.56482899299555443109336624908, 2.49561686725426978994739439666, 3.34429622718569908729876339985, 3.80527778884990097102940261143, 5.19936486244070317479880685098, 5.66146745069945851023919856596, 6.41777743316307942878589698341, 7.42070894767394719178986604652, 7.80564621191749223666525872411
