| L(s) = 1 | − 5-s + 11-s − 6·13-s + 6·17-s + 25-s + 6·29-s − 8·31-s − 2·37-s + 2·41-s + 4·43-s − 7·49-s − 6·53-s − 55-s + 4·59-s − 10·61-s + 6·65-s + 12·67-s − 10·73-s + 12·79-s − 6·85-s + 6·89-s + 10·97-s − 10·101-s + 16·103-s − 8·107-s − 10·109-s − 2·113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.66·13-s + 1.45·17-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.134·55-s + 0.520·59-s − 1.28·61-s + 0.744·65-s + 1.46·67-s − 1.17·73-s + 1.35·79-s − 0.650·85-s + 0.635·89-s + 1.01·97-s − 0.995·101-s + 1.57·103-s − 0.773·107-s − 0.957·109-s − 0.188·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.62188393944448843765133599601, −6.92618582727511893386818312575, −6.14619152253662608191194086863, −5.23550808922635776474874174435, −4.80394327096564033262825653141, −3.85048478865134550152649816832, −3.14653241795232383848891002499, −2.30778633812731160166566149334, −1.20295562616902398643985488144, 0,
1.20295562616902398643985488144, 2.30778633812731160166566149334, 3.14653241795232383848891002499, 3.85048478865134550152649816832, 4.80394327096564033262825653141, 5.23550808922635776474874174435, 6.14619152253662608191194086863, 6.92618582727511893386818312575, 7.62188393944448843765133599601
