| L(s) = 1 | − 4.38·5-s − 7-s + 5.38·11-s − 2.54·13-s + 2.58·17-s − 6.72·19-s + 0.800·23-s + 14.1·25-s − 3.74·29-s + 3.39·31-s + 4.38·35-s + 4.38·37-s + 6.39·41-s − 0.763·43-s + 8.26·47-s + 49-s + 4.94·53-s − 23.5·55-s − 5.57·59-s − 8.28·61-s + 11.1·65-s − 1.89·67-s − 1.34·71-s − 8.65·73-s − 5.38·77-s − 13.2·79-s − 3.72·83-s + ⋯ |
| L(s) = 1 | − 1.95·5-s − 0.377·7-s + 1.62·11-s − 0.705·13-s + 0.625·17-s − 1.54·19-s + 0.166·23-s + 2.83·25-s − 0.695·29-s + 0.609·31-s + 0.740·35-s + 0.720·37-s + 0.998·41-s − 0.116·43-s + 1.20·47-s + 0.142·49-s + 0.679·53-s − 3.17·55-s − 0.725·59-s − 1.06·61-s + 1.38·65-s − 0.231·67-s − 0.159·71-s − 1.01·73-s − 0.613·77-s − 1.49·79-s − 0.408·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 5 | \( 1 + 4.38T + 5T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6.72T + 19T^{2} \) |
| 23 | \( 1 - 0.800T + 23T^{2} \) |
| 29 | \( 1 + 3.74T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 - 4.38T + 37T^{2} \) |
| 41 | \( 1 - 6.39T + 41T^{2} \) |
| 43 | \( 1 + 0.763T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 + 8.28T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 + 1.34T + 71T^{2} \) |
| 73 | \( 1 + 8.65T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 3.72T + 83T^{2} \) |
| 89 | \( 1 - 6.99T + 89T^{2} \) |
| 97 | \( 1 - 2.96T + 97T^{2} \) |
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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
−7.82600319168038491627214971440, −7.35720645663727663464049080945, −6.66401024047855663803693715972, −5.93088538793161958815861636101, −4.59460910475134648917370295440, −4.18483921558356449308893635970, −3.56503231456837626234597641477, −2.63413624953025496072845156526, −1.12721958478017091248228600568, 0,
1.12721958478017091248228600568, 2.63413624953025496072845156526, 3.56503231456837626234597641477, 4.18483921558356449308893635970, 4.59460910475134648917370295440, 5.93088538793161958815861636101, 6.66401024047855663803693715972, 7.35720645663727663464049080945, 7.82600319168038491627214971440
