| L(s) = 1 | − 3-s + 5-s + 2.89·7-s + 9-s − 1.27·11-s + 0.474·13-s − 15-s − 3.35·17-s − 3.09·19-s − 2.89·21-s + 2.70·23-s + 25-s − 27-s − 5.29·29-s − 10.8·31-s + 1.27·33-s + 2.89·35-s − 8.93·37-s − 0.474·39-s − 2.00·41-s − 5.48·43-s + 45-s + 2.02·47-s + 1.39·49-s + 3.35·51-s − 7.07·53-s − 1.27·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.09·7-s + 0.333·9-s − 0.384·11-s + 0.131·13-s − 0.258·15-s − 0.813·17-s − 0.709·19-s − 0.632·21-s + 0.564·23-s + 0.200·25-s − 0.192·27-s − 0.983·29-s − 1.95·31-s + 0.222·33-s + 0.489·35-s − 1.46·37-s − 0.0759·39-s − 0.313·41-s − 0.837·43-s + 0.149·45-s + 0.295·47-s + 0.198·49-s + 0.469·51-s − 0.972·53-s − 0.172·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 13 | \( 1 - 0.474T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 + 2.00T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 - 2.02T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 4.68T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 7.77T + 89T^{2} \) |
| 97 | \( 1 + 7.67T + 97T^{2} \) |
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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
−8.136760067294057889972692696423, −7.20077109970624886421657773229, −6.70106808555123635142182809062, −5.62507138397169845978467703352, −5.22557063152331660654879505428, −4.44474151737620504083229664768, −3.51675200390176891100692193003, −2.16232974100186983645419391827, −1.56666661502085449972045843761, 0,
1.56666661502085449972045843761, 2.16232974100186983645419391827, 3.51675200390176891100692193003, 4.44474151737620504083229664768, 5.22557063152331660654879505428, 5.62507138397169845978467703352, 6.70106808555123635142182809062, 7.20077109970624886421657773229, 8.136760067294057889972692696423
