| L(s) = 1 | + 2.34·3-s − 1.00·5-s − 0.336·7-s + 2.50·9-s − 1.49·11-s + 2.73·13-s − 2.36·15-s − 1.85·17-s − 7.68·19-s − 0.790·21-s − 3.98·25-s − 1.15·27-s − 4.19·29-s + 4.98·31-s − 3.50·33-s + 0.338·35-s − 10.7·37-s + 6.41·39-s − 2.63·41-s + 3.44·43-s − 2.52·45-s − 4.41·47-s − 6.88·49-s − 4.34·51-s + 6.26·53-s + 1.50·55-s − 18.0·57-s + ⋯ |
| L(s) = 1 | + 1.35·3-s − 0.449·5-s − 0.127·7-s + 0.836·9-s − 0.449·11-s + 0.758·13-s − 0.609·15-s − 0.449·17-s − 1.76·19-s − 0.172·21-s − 0.797·25-s − 0.221·27-s − 0.778·29-s + 0.895·31-s − 0.609·33-s + 0.0572·35-s − 1.76·37-s + 1.02·39-s − 0.411·41-s + 0.524·43-s − 0.376·45-s − 0.643·47-s − 0.983·49-s − 0.609·51-s + 0.861·53-s + 0.202·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 + 0.336T + 7T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 2.63T + 41T^{2} \) |
| 43 | \( 1 - 3.44T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 - 6.26T + 53T^{2} \) |
| 59 | \( 1 - 4.19T + 59T^{2} \) |
| 61 | \( 1 + 4.96T + 61T^{2} \) |
| 67 | \( 1 - 4.20T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.228375205497130933504680398163, −7.52914942004487009674926091839, −6.68930233801041468438669646413, −5.94774975680767141875118679578, −4.82842042084086655375214633494, −3.93665559296949271208139482296, −3.46733561216773114238390170581, −2.46208172240475844516707171206, −1.76209637763590367342151347497, 0,
1.76209637763590367342151347497, 2.46208172240475844516707171206, 3.46733561216773114238390170581, 3.93665559296949271208139482296, 4.82842042084086655375214633494, 5.94774975680767141875118679578, 6.68930233801041468438669646413, 7.52914942004487009674926091839, 8.228375205497130933504680398163
