| L(s) = 1 | − 3.23·3-s − 3.23·5-s − 7-s + 7.47·9-s − 4·11-s + 3.23·13-s + 10.4·15-s − 2·17-s + 3.23·19-s + 3.23·21-s + 2.47·23-s + 5.47·25-s − 14.4·27-s + 10.4·29-s + 12.9·33-s + 3.23·35-s − 2.47·37-s − 10.4·39-s − 2·41-s + 8.94·43-s − 24.1·45-s − 4.94·47-s + 49-s + 6.47·51-s + 8.94·53-s + 12.9·55-s − 10.4·57-s + ⋯ |
| L(s) = 1 | − 1.86·3-s − 1.44·5-s − 0.377·7-s + 2.49·9-s − 1.20·11-s + 0.897·13-s + 2.70·15-s − 0.485·17-s + 0.742·19-s + 0.706·21-s + 0.515·23-s + 1.09·25-s − 2.78·27-s + 1.94·29-s + 2.25·33-s + 0.546·35-s − 0.406·37-s − 1.67·39-s − 0.312·41-s + 1.36·43-s − 3.60·45-s − 0.721·47-s + 0.142·49-s + 0.906·51-s + 1.22·53-s + 1.74·55-s − 1.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 + 3.23T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 4.76T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−8.808368632162736066202741210694, −7.86160844803531547269126701210, −7.18990165055049598164849563608, −6.46441875304653880966488143750, −5.62483651115504444349883694516, −4.80711609506956019436257480220, −4.15730552800527270694336163481, −3.03429556676906260884683459961, −1.02333097575618114958165350669, 0,
1.02333097575618114958165350669, 3.03429556676906260884683459961, 4.15730552800527270694336163481, 4.80711609506956019436257480220, 5.62483651115504444349883694516, 6.46441875304653880966488143750, 7.18990165055049598164849563608, 7.86160844803531547269126701210, 8.808368632162736066202741210694
