| L(s) = 1 | + 4.18·5-s − 3i·11-s + 2.44i·13-s − 1.01·17-s − 1.01i·19-s + 4.24i·23-s + 12.4·25-s − 1.24i·29-s − 5.61i·31-s + 8.24·37-s − 2.02·41-s − 8.24·43-s + 1.01·47-s − 1.24i·53-s − 12.5i·55-s + ⋯ |
| L(s) = 1 | + 1.87·5-s − 0.904i·11-s + 0.679i·13-s − 0.246·17-s − 0.232i·19-s + 0.884i·23-s + 2.49·25-s − 0.230i·29-s − 1.00i·31-s + 1.35·37-s − 0.316·41-s − 1.25·43-s + 0.147·47-s − 0.170i·53-s − 1.69i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.133979739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.133979739\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.18T + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 + 1.01iT - 19T^{2} \) |
| 23 | \( 1 - 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 1.24iT - 29T^{2} \) |
| 31 | \( 1 + 5.61iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 - 1.01T + 47T^{2} \) |
| 53 | \( 1 + 1.24iT - 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 5.91iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 10.2iT - 71T^{2} \) |
| 73 | \( 1 - 8.36iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 3.76iT - 97T^{2} \) |
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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
−8.052989080448739214274479362701, −6.93146573186517844811994144445, −6.48787593832010308235527252931, −5.74335504425538839823473455910, −5.35871563286590220244288819501, −4.42229457691781453054362039907, −3.42340221267166222053804507053, −2.48211103233582094895701255927, −1.88397087121590489365828848818, −0.873636422810191570444985557342,
0.988785527597009818503995775931, 1.97452190343926226514562344643, 2.50185181514094788940795027570, 3.45536450607591968753137663720, 4.71332805094586414666478606650, 5.10651485679254785750903457185, 5.92625746528727610487238905306, 6.49956418864013524356902965970, 7.06560378187975161619959127773, 8.033199043830169285945352358638
