| L(s) = 1 | − 3-s − 5-s + 3.41·7-s + 9-s − 2.82·11-s − 5.41·13-s + 15-s − 4·17-s + 2.82·19-s − 3.41·21-s + 6·23-s + 25-s − 27-s − 6.24·29-s − 31-s + 2.82·33-s − 3.41·35-s + 1.41·37-s + 5.41·39-s + 4.82·41-s − 11.3·43-s − 45-s − 0.828·47-s + 4.65·49-s + 4·51-s − 4·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.29·7-s + 0.333·9-s − 0.852·11-s − 1.50·13-s + 0.258·15-s − 0.970·17-s + 0.648·19-s − 0.745·21-s + 1.25·23-s + 0.200·25-s − 0.192·27-s − 1.15·29-s − 0.179·31-s + 0.492·33-s − 0.577·35-s + 0.232·37-s + 0.866·39-s + 0.754·41-s − 1.72·43-s − 0.149·45-s − 0.120·47-s + 0.665·49-s + 0.560·51-s − 0.549·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.167309033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.167309033\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 9.07T + 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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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
−7.85109625383074091524530253886, −7.23570002974757639498738593370, −6.70065373726535365261827172518, −5.50744895421640275081135967646, −4.94871140419464730161662075171, −4.72883425235970392775767471500, −3.62945974178702252387346971145, −2.56143809384325590381760667340, −1.80753270361967274804468633978, −0.54961145242667611209704256887,
0.54961145242667611209704256887, 1.80753270361967274804468633978, 2.56143809384325590381760667340, 3.62945974178702252387346971145, 4.72883425235970392775767471500, 4.94871140419464730161662075171, 5.50744895421640275081135967646, 6.70065373726535365261827172518, 7.23570002974757639498738593370, 7.85109625383074091524530253886
