| L(s) = 1 | + 5-s − 3·11-s − 13-s + 2·17-s − 4·19-s − 4·23-s + 25-s − 4·29-s + 3·37-s + 2·41-s − 2·43-s + 6·47-s + 13·53-s − 3·55-s + 5·59-s − 7·61-s − 65-s − 13·67-s − 5·71-s + 11·73-s + 13·79-s + 4·83-s + 2·85-s − 89-s − 4·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.904·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.742·29-s + 0.493·37-s + 0.312·41-s − 0.304·43-s + 0.875·47-s + 1.78·53-s − 0.404·55-s + 0.650·59-s − 0.896·61-s − 0.124·65-s − 1.58·67-s − 0.593·71-s + 1.28·73-s + 1.46·79-s + 0.439·83-s + 0.216·85-s − 0.105·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−13.31198120772778, −12.56904444041517, −12.37008991622771, −11.86607274989935, −11.24501256535840, −10.70915768942471, −10.40066807881437, −10.00848470627672, −9.376822830674126, −9.069864523419398, −8.402255574381136, −7.929678979339541, −7.607216279497511, −6.954413295723547, −6.497018187461492, −5.819200589193665, −5.561162820433586, −5.049136227534216, −4.305829551694497, −3.988814450072547, −3.229986967468139, −2.579056293431334, −2.212597878053729, −1.583796787318993, −0.7326772163901885, 0,
0.7326772163901885, 1.583796787318993, 2.212597878053729, 2.579056293431334, 3.229986967468139, 3.988814450072547, 4.305829551694497, 5.049136227534216, 5.561162820433586, 5.819200589193665, 6.497018187461492, 6.954413295723547, 7.607216279497511, 7.929678979339541, 8.402255574381136, 9.069864523419398, 9.376822830674126, 10.00848470627672, 10.40066807881437, 10.70915768942471, 11.24501256535840, 11.86607274989935, 12.37008991622771, 12.56904444041517, 13.31198120772778
