| L(s) = 1 | + 3-s + 2.56·5-s − 3.12·7-s + 9-s + 6.56·11-s + 0.561·13-s + 2.56·15-s − 7.68·19-s − 3.12·21-s + 3.43·23-s + 1.56·25-s + 27-s − 1.12·29-s + 8.24·31-s + 6.56·33-s − 8·35-s + 4·37-s + 0.561·39-s − 9.68·41-s + 7.68·43-s + 2.56·45-s − 9.12·47-s + 2.75·49-s + 6·53-s + 16.8·55-s − 7.68·57-s + 11.3·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.14·5-s − 1.18·7-s + 0.333·9-s + 1.97·11-s + 0.155·13-s + 0.661·15-s − 1.76·19-s − 0.681·21-s + 0.716·23-s + 0.312·25-s + 0.192·27-s − 0.208·29-s + 1.48·31-s + 1.14·33-s − 1.35·35-s + 0.657·37-s + 0.0899·39-s − 1.51·41-s + 1.17·43-s + 0.381·45-s − 1.33·47-s + 0.393·49-s + 0.824·53-s + 2.26·55-s − 1.01·57-s + 1.48·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.230647903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.230647903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 6.56T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 - 0.876T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−8.195983584576162435390916806028, −6.85119828518875261172998987479, −6.54602978386741404896136813085, −6.21554219258263399941526679773, −5.15903378188724527284977241025, −4.10580939796266996357410251039, −3.64223349972927451300291855392, −2.63300341860595358636316291710, −1.92748467138352155135275064673, −0.920166081804844074340831556326,
0.920166081804844074340831556326, 1.92748467138352155135275064673, 2.63300341860595358636316291710, 3.64223349972927451300291855392, 4.10580939796266996357410251039, 5.15903378188724527284977241025, 6.21554219258263399941526679773, 6.54602978386741404896136813085, 6.85119828518875261172998987479, 8.195983584576162435390916806028
