| L(s) = 1 | − 3·5-s + 3·11-s + 2·13-s − 3·17-s − 19-s − 3·23-s + 4·25-s + 6·29-s − 7·31-s − 37-s − 6·41-s − 4·43-s + 9·47-s − 3·53-s − 9·55-s − 9·59-s − 61-s − 6·65-s − 7·67-s − 73-s − 13·79-s − 12·83-s + 9·85-s − 15·89-s + 3·95-s − 10·97-s − 15·101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.904·11-s + 0.554·13-s − 0.727·17-s − 0.229·19-s − 0.625·23-s + 4/5·25-s + 1.11·29-s − 1.25·31-s − 0.164·37-s − 0.937·41-s − 0.609·43-s + 1.31·47-s − 0.412·53-s − 1.21·55-s − 1.17·59-s − 0.128·61-s − 0.744·65-s − 0.855·67-s − 0.117·73-s − 1.46·79-s − 1.31·83-s + 0.976·85-s − 1.58·89-s + 0.307·95-s − 1.01·97-s − 1.49·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.690932609266128059888188385854, −8.262637018939911096603858847029, −7.27657735328498763814943158038, −6.67413798525527807153917855846, −5.71711213824960626307579454428, −4.44529829557613531132250015524, −3.98660375345473223094112675622, −3.04451713874417289677681835886, −1.54505867526772913791014367907, 0,
1.54505867526772913791014367907, 3.04451713874417289677681835886, 3.98660375345473223094112675622, 4.44529829557613531132250015524, 5.71711213824960626307579454428, 6.67413798525527807153917855846, 7.27657735328498763814943158038, 8.262637018939911096603858847029, 8.690932609266128059888188385854
