| L(s) = 1 | − 3-s − 3·5-s + 7-s − 2·9-s + 11-s + 4·13-s + 3·15-s − 6·17-s − 2·19-s − 21-s + 3·23-s + 4·25-s + 5·27-s + 6·29-s + 5·31-s − 33-s − 3·35-s − 11·37-s − 4·39-s + 6·41-s − 8·43-s + 6·45-s + 49-s + 6·51-s + 6·53-s − 3·55-s + 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 1.10·13-s + 0.774·15-s − 1.45·17-s − 0.458·19-s − 0.218·21-s + 0.625·23-s + 4/5·25-s + 0.962·27-s + 1.11·29-s + 0.898·31-s − 0.174·33-s − 0.507·35-s − 1.80·37-s − 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.894·45-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.404·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.154848181833606301816411535200, −6.89635932514915681940214174888, −6.69227362449418376626075160667, −5.70008544609539374844814353462, −4.84767148577891784015091892690, −4.20086015444217019855350266017, −3.48528504268689712306029643361, −2.47348180702840979174904080971, −1.08054599527764648143837096247, 0,
1.08054599527764648143837096247, 2.47348180702840979174904080971, 3.48528504268689712306029643361, 4.20086015444217019855350266017, 4.84767148577891784015091892690, 5.70008544609539374844814353462, 6.69227362449418376626075160667, 6.89635932514915681940214174888, 8.154848181833606301816411535200
