| L(s) = 1 | + 1.58i·5-s − 7-s + 0.790i·11-s − 0.494i·13-s + 4.46·17-s + 7.55i·19-s − 0.839·23-s + 2.48·25-s + 2.76i·29-s + 0.568·31-s − 1.58i·35-s − 0.343i·37-s − 5.93·41-s + 3.16i·43-s − 4.80·47-s + ⋯ |
| L(s) = 1 | + 0.708i·5-s − 0.377·7-s + 0.238i·11-s − 0.137i·13-s + 1.08·17-s + 1.73i·19-s − 0.175·23-s + 0.497·25-s + 0.513i·29-s + 0.102·31-s − 0.267i·35-s − 0.0564i·37-s − 0.926·41-s + 0.482i·43-s − 0.701·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.360999351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.360999351\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.58iT - 5T^{2} \) |
| 11 | \( 1 - 0.790iT - 11T^{2} \) |
| 13 | \( 1 + 0.494iT - 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 - 7.55iT - 19T^{2} \) |
| 23 | \( 1 + 0.839T + 23T^{2} \) |
| 29 | \( 1 - 2.76iT - 29T^{2} \) |
| 31 | \( 1 - 0.568T + 31T^{2} \) |
| 37 | \( 1 + 0.343iT - 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 - 3.16iT - 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 + 4.36iT - 53T^{2} \) |
| 59 | \( 1 - 4.50iT - 59T^{2} \) |
| 61 | \( 1 + 5.40iT - 61T^{2} \) |
| 67 | \( 1 + 7.57iT - 67T^{2} \) |
| 71 | \( 1 + 9.52T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 + 7.30iT - 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 1.75T + 97T^{2} \) |
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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
−8.061968380400983756298054834172, −7.75696254442732443941457674197, −6.81365839684934310573334030995, −6.30520286168339312371748200997, −5.53908562706326753610164436402, −4.79041386169508348412737766649, −3.57587597542098428718871123091, −3.34649766751908191419147692437, −2.22509897252911133388539928516, −1.21975756565595026170482591512,
0.37406538348966016161243375504, 1.32285870171113252277626686413, 2.54530190106562281437942808951, 3.29928709468683799842786454775, 4.23093324754405344933021938992, 4.99450977563813420806454527380, 5.55295982899430073161705123771, 6.47502342719813746591443630788, 7.06113812241011393698200776936, 7.891003122224968226302299990009
