| L(s) = 1 | + 2.10·3-s − 2.97·5-s + 1.34·7-s + 1.44·9-s + 11-s − 3.44·13-s − 6.26·15-s + 4.36·17-s − 19-s + 2.83·21-s − 8.16·23-s + 3.82·25-s − 3.27·27-s − 1.36·29-s − 2.69·31-s + 2.10·33-s − 3.99·35-s − 1.17·37-s − 7.26·39-s − 3.36·41-s + 3.18·43-s − 4.29·45-s − 6.44·47-s − 5.19·49-s + 9.21·51-s + 9.21·53-s − 2.97·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s − 1.32·5-s + 0.508·7-s + 0.482·9-s + 0.301·11-s − 0.955·13-s − 1.61·15-s + 1.05·17-s − 0.229·19-s + 0.618·21-s − 1.70·23-s + 0.765·25-s − 0.630·27-s − 0.252·29-s − 0.484·31-s + 0.367·33-s − 0.675·35-s − 0.192·37-s − 1.16·39-s − 0.525·41-s + 0.486·43-s − 0.640·45-s − 0.940·47-s − 0.741·49-s + 1.29·51-s + 1.26·53-s − 0.400·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 + 2.97T + 5T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 23 | \( 1 + 8.16T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 2.69T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 9.21T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 - 0.802T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 2.41T + 71T^{2} \) |
| 73 | \( 1 + 2.83T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 - 8.70T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.243069911069036063558547350902, −7.60471469746022126578166104846, −7.26867678463507376853929594747, −5.99488302308658565869312276011, −4.99034071762546591725659336774, −4.05352745777888575307352246959, −3.60044522559543775777636633338, −2.66357623335323196513486484207, −1.67666416403526608824822246126, 0,
1.67666416403526608824822246126, 2.66357623335323196513486484207, 3.60044522559543775777636633338, 4.05352745777888575307352246959, 4.99034071762546591725659336774, 5.99488302308658565869312276011, 7.26867678463507376853929594747, 7.60471469746022126578166104846, 8.243069911069036063558547350902
