| L(s) = 1 | + 3.41i·5-s + 0.585·7-s − 2i·11-s − 2.82i·13-s + 7.65·17-s − 5.65i·19-s − 6.82·23-s − 6.65·25-s − 3.41i·29-s − 7.41·31-s + 2i·35-s − 1.65i·37-s − 0.343·41-s − 9.65i·43-s − 4.48·47-s + ⋯ |
| L(s) = 1 | + 1.52i·5-s + 0.221·7-s − 0.603i·11-s − 0.784i·13-s + 1.85·17-s − 1.29i·19-s − 1.42·23-s − 1.33·25-s − 0.634i·29-s − 1.33·31-s + 0.338i·35-s − 0.272i·37-s − 0.0535·41-s − 1.47i·43-s − 0.654·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040261811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040261811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.41iT - 5T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 + 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 + 3.41iT - 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 + 1.65iT - 37T^{2} \) |
| 41 | \( 1 + 0.343T + 41T^{2} \) |
| 43 | \( 1 + 9.65iT - 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 7.89iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 1.65iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−7.968286522095441317591790827302, −7.40059968273424425885742643519, −6.81431002346493054622580990997, −5.75968445501680270211203448855, −5.58396203307732431183139088584, −4.21729156958271808481788530140, −3.30481885441278746030408684627, −2.90882498536768087829469425631, −1.80003910916172094604559066927, −0.27711565706114710016875860092,
1.38929582007157612389015645166, 1.70647320274923551760404857302, 3.28572263298868847782614675995, 4.14907299330438271386954608392, 4.75387054156745692400070625279, 5.57031318318754740468010663955, 6.05789979171117677786025044866, 7.26054574348685493102133065803, 7.957183912395862851880549298396, 8.336746563082434048668877742592
