| L(s) = 1 | + 2·3-s + 5-s + 9-s + 2.27·11-s − 2·13-s + 2·15-s − 7.27·17-s + 19-s − 7.54·23-s − 4·25-s − 4·27-s − 4·29-s + 4·31-s + 4.54·33-s + 2·37-s − 4·39-s + 4.54·41-s − 43-s + 45-s + 2.27·47-s − 14.5·51-s − 6.54·53-s + 2.27·55-s + 2·57-s + 4·59-s − 12.2·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.333·9-s + 0.685·11-s − 0.554·13-s + 0.516·15-s − 1.76·17-s + 0.229·19-s − 1.57·23-s − 0.800·25-s − 0.769·27-s − 0.742·29-s + 0.718·31-s + 0.792·33-s + 0.328·37-s − 0.640·39-s + 0.710·41-s − 0.152·43-s + 0.149·45-s + 0.331·47-s − 2.03·51-s − 0.899·53-s + 0.306·55-s + 0.264·57-s + 0.520·59-s − 1.57·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 23 | \( 1 + 7.54T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 2.27T + 47T^{2} \) |
| 53 | \( 1 + 6.54T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 - 0.549T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 8.54T + 97T^{2} \) |
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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
−7.71714917823861638993109811044, −6.90859687520260318448077582540, −6.20283407147173623240099769954, −5.55528799533582143705735827223, −4.35780900733287766682040458910, −4.03452564750507730897093142648, −2.97466419594173721666056639922, −2.25742215999456096568911350087, −1.68847621078451017832376543411, 0,
1.68847621078451017832376543411, 2.25742215999456096568911350087, 2.97466419594173721666056639922, 4.03452564750507730897093142648, 4.35780900733287766682040458910, 5.55528799533582143705735827223, 6.20283407147173623240099769954, 6.90859687520260318448077582540, 7.71714917823861638993109811044
