| L(s) = 1 | − 4·7-s − 6·11-s + 13-s − 6·17-s + 6·19-s + 2·23-s − 2·29-s − 2·31-s + 10·37-s − 10·41-s − 6·43-s + 4·47-s + 9·49-s − 6·53-s − 10·59-s − 14·61-s + 8·67-s − 6·71-s + 6·73-s + 24·77-s − 12·79-s − 4·83-s − 10·89-s − 4·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.80·11-s + 0.277·13-s − 1.45·17-s + 1.37·19-s + 0.417·23-s − 0.371·29-s − 0.359·31-s + 1.64·37-s − 1.56·41-s − 0.914·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s − 1.30·59-s − 1.79·61-s + 0.977·67-s − 0.712·71-s + 0.702·73-s + 2.73·77-s − 1.35·79-s − 0.439·83-s − 1.05·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.06227008545153, −13.69781239610853, −13.24553247471699, −12.92253826914681, −12.60866903514677, −11.86232949707017, −11.26223459438040, −10.89396449829947, −10.28419563058074, −9.889285039776442, −9.363965336171394, −8.984867399062850, −8.258891156108736, −7.778864743697702, −7.174597143355081, −6.799755118961392, −6.078057958189529, −5.753471108100729, −5.005256066890036, −4.610311749108721, −3.753265849843517, −3.060947241426661, −2.881391944702174, −2.133011747521608, −1.209376002783732, 0, 0,
1.209376002783732, 2.133011747521608, 2.881391944702174, 3.060947241426661, 3.753265849843517, 4.610311749108721, 5.005256066890036, 5.753471108100729, 6.078057958189529, 6.799755118961392, 7.174597143355081, 7.778864743697702, 8.258891156108736, 8.984867399062850, 9.363965336171394, 9.889285039776442, 10.28419563058074, 10.89396449829947, 11.26223459438040, 11.86232949707017, 12.60866903514677, 12.92253826914681, 13.24553247471699, 13.69781239610853, 14.06227008545153
