| L(s) = 1 | − 3-s − 5-s + 9-s − 2·11-s + 13-s + 15-s + 3·19-s + 25-s − 27-s − 9·31-s + 2·33-s + 3·37-s − 39-s + 2·41-s + 3·43-s − 45-s − 6·47-s + 2·55-s − 3·57-s − 4·59-s + 2·61-s − 65-s + 5·67-s + 14·71-s + 73-s − 75-s + 9·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.258·15-s + 0.688·19-s + 1/5·25-s − 0.192·27-s − 1.61·31-s + 0.348·33-s + 0.493·37-s − 0.160·39-s + 0.312·41-s + 0.457·43-s − 0.149·45-s − 0.875·47-s + 0.269·55-s − 0.397·57-s − 0.520·59-s + 0.256·61-s − 0.124·65-s + 0.610·67-s + 1.66·71-s + 0.117·73-s − 0.115·75-s + 1.01·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.69586370017970350333227670404, −7.08512218635799206865759704980, −6.30739452241626198565634149819, −5.50019399355657193225966745888, −4.99383628205177838209295559109, −4.05292016300874672019716278717, −3.35661215618630392947433847763, −2.32026952118503041326962166819, −1.16595590154436500890254318643, 0,
1.16595590154436500890254318643, 2.32026952118503041326962166819, 3.35661215618630392947433847763, 4.05292016300874672019716278717, 4.99383628205177838209295559109, 5.50019399355657193225966745888, 6.30739452241626198565634149819, 7.08512218635799206865759704980, 7.69586370017970350333227670404
