| L(s) = 1 | + 1.88·3-s + 2.88·5-s − 4.36·7-s + 0.537·9-s + 5.57·11-s − 6.28·13-s + 5.42·15-s − 3.43·17-s + 19-s − 8.21·21-s − 8.88·23-s + 3.31·25-s − 4.63·27-s − 0.842·29-s − 10.4·31-s + 10.4·33-s − 12.5·35-s − 11.5·37-s − 11.8·39-s + 3.86·41-s − 11.6·43-s + 1.54·45-s + 3.21·47-s + 12.0·49-s − 6.46·51-s − 53-s + 16.0·55-s + ⋯ |
| L(s) = 1 | + 1.08·3-s + 1.28·5-s − 1.65·7-s + 0.179·9-s + 1.67·11-s − 1.74·13-s + 1.40·15-s − 0.833·17-s + 0.229·19-s − 1.79·21-s − 1.85·23-s + 0.663·25-s − 0.891·27-s − 0.156·29-s − 1.86·31-s + 1.82·33-s − 2.12·35-s − 1.89·37-s − 1.89·39-s + 0.603·41-s − 1.77·43-s + 0.231·45-s + 0.468·47-s + 1.72·49-s − 0.904·51-s − 0.137·53-s + 2.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 1.88T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 11 | \( 1 - 5.57T + 11T^{2} \) |
| 13 | \( 1 + 6.28T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 23 | \( 1 + 8.88T + 23T^{2} \) |
| 29 | \( 1 + 0.842T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 3.21T + 47T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 9.50T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 6.20T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 4.97T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.258350910616023200767730526555, −7.10970548156393113271688125859, −6.72443459746686808259944318455, −5.98612113983804490735420569936, −5.20734451459607983766379814396, −3.84693603302941964750704177939, −3.46013796265091473048163795374, −2.24309302238258947904965152718, −1.99775947291185451001007723667, 0,
1.99775947291185451001007723667, 2.24309302238258947904965152718, 3.46013796265091473048163795374, 3.84693603302941964750704177939, 5.20734451459607983766379814396, 5.98612113983804490735420569936, 6.72443459746686808259944318455, 7.10970548156393113271688125859, 8.258350910616023200767730526555
