| L(s) = 1 | + 2.28·5-s − 7-s − 2.82·11-s − 3.23·13-s + 0.540·17-s + 19-s + 2.82·23-s + 0.236·25-s − 8.61·29-s + 7.70·31-s − 2.28·35-s + 4.47·37-s − 8.48·41-s − 5.23·43-s − 4.03·47-s + 49-s + 11.4·53-s − 6.47·55-s + 2.16·59-s − 3.52·61-s − 7.40·65-s − 15.4·67-s + 12.5·71-s − 10.9·73-s + 2.82·77-s − 10.4·79-s − 7.94·83-s + ⋯ |
| L(s) = 1 | + 1.02·5-s − 0.377·7-s − 0.852·11-s − 0.897·13-s + 0.131·17-s + 0.229·19-s + 0.589·23-s + 0.0472·25-s − 1.59·29-s + 1.38·31-s − 0.386·35-s + 0.735·37-s − 1.32·41-s − 0.798·43-s − 0.588·47-s + 0.142·49-s + 1.57·53-s − 0.872·55-s + 0.281·59-s − 0.451·61-s − 0.918·65-s − 1.88·67-s + 1.48·71-s − 1.28·73-s + 0.322·77-s − 1.17·79-s − 0.872·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.28T + 5T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 0.540T + 17T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 8.61T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.94T + 83T^{2} \) |
| 89 | \( 1 + 8.48T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.86241086766160016088874873561, −7.17873846115709772546149403634, −6.46415764038948065019244694139, −5.59888776590281603955370860073, −5.20394621947103789740234746554, −4.25152600819746338855200793057, −3.08182528017810445194770658421, −2.47338784824763596029298995886, −1.50109432184343539127572684278, 0,
1.50109432184343539127572684278, 2.47338784824763596029298995886, 3.08182528017810445194770658421, 4.25152600819746338855200793057, 5.20394621947103789740234746554, 5.59888776590281603955370860073, 6.46415764038948065019244694139, 7.17873846115709772546149403634, 7.86241086766160016088874873561
