| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 6·13-s − 15-s − 2·17-s − 4·19-s + 25-s + 27-s − 10·29-s − 4·31-s + 4·33-s + 10·37-s + 6·39-s − 2·41-s − 4·43-s − 45-s + 8·47-s − 2·51-s − 2·53-s − 4·55-s − 4·57-s − 12·59-s − 10·61-s − 6·65-s + 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.696·33-s + 1.64·37-s + 0.960·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s − 0.274·53-s − 0.539·55-s − 0.529·57-s − 1.56·59-s − 1.28·61-s − 0.744·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.77937763785852, −14.43981146347493, −13.83418836494610, −13.22750741871504, −12.98257634495107, −12.34828989246636, −11.63585609165121, −11.19078437407265, −10.87930046792038, −10.20906514658380, −9.339614107311593, −9.012235895833833, −8.749256835308109, −7.920450080431045, −7.598798145308115, −6.815171242051615, −6.246469649443073, −5.944891438020776, −4.955923157418843, −4.159655202371246, −3.867853919360280, −3.405173793839786, −2.483778266198529, −1.691638444009990, −1.161529993198741, 0,
1.161529993198741, 1.691638444009990, 2.483778266198529, 3.405173793839786, 3.867853919360280, 4.159655202371246, 4.955923157418843, 5.944891438020776, 6.246469649443073, 6.815171242051615, 7.598798145308115, 7.920450080431045, 8.749256835308109, 9.012235895833833, 9.339614107311593, 10.20906514658380, 10.87930046792038, 11.19078437407265, 11.63585609165121, 12.34828989246636, 12.98257634495107, 13.22750741871504, 13.83418836494610, 14.43981146347493, 14.77937763785852
