| L(s) = 1 | − 3-s + 2·5-s + 4·7-s − 2·9-s − 2·11-s − 7·13-s − 2·15-s − 4·17-s − 6·19-s − 4·21-s + 23-s − 25-s + 5·27-s − 5·29-s − 3·31-s + 2·33-s + 8·35-s − 2·37-s + 7·39-s − 9·41-s + 8·43-s − 4·45-s + 47-s + 9·49-s + 4·51-s + 6·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s − 2/3·9-s − 0.603·11-s − 1.94·13-s − 0.516·15-s − 0.970·17-s − 1.37·19-s − 0.872·21-s + 0.208·23-s − 1/5·25-s + 0.962·27-s − 0.928·29-s − 0.538·31-s + 0.348·33-s + 1.35·35-s − 0.328·37-s + 1.12·39-s − 1.40·41-s + 1.21·43-s − 0.596·45-s + 0.145·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.071741990646530995799311700064, −8.336842801922936941253367668238, −7.48334861463485721195009336896, −6.61508499461346018061752445938, −5.50380083802215924411826393438, −5.15358677066922271085410464002, −4.31313200192667428557367021045, −2.46164952980548405624442154475, −1.97056330740558525387471188060, 0,
1.97056330740558525387471188060, 2.46164952980548405624442154475, 4.31313200192667428557367021045, 5.15358677066922271085410464002, 5.50380083802215924411826393438, 6.61508499461346018061752445938, 7.48334861463485721195009336896, 8.336842801922936941253367668238, 9.071741990646530995799311700064
