| L(s) = 1 | − 2.21·5-s − 1.10·7-s − 1.52i·11-s + 3.26i·13-s − 5.68·17-s + 6.25·19-s + 4.86·23-s − 0.0972·25-s − 8.93i·29-s − 1.89·31-s + 2.45·35-s + 4.41i·37-s − 2.75·41-s + 0.818i·43-s + 3.00i·47-s + ⋯ |
| L(s) = 1 | − 0.990·5-s − 0.418·7-s − 0.460i·11-s + 0.905i·13-s − 1.37·17-s + 1.43·19-s + 1.01·23-s − 0.0194·25-s − 1.65i·29-s − 0.340·31-s + 0.414·35-s + 0.725i·37-s − 0.429·41-s + 0.124i·43-s + 0.438i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092248451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092248451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + (8.71 + 9.53i)T \) |
| good | 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 11 | \( 1 + 1.52iT - 11T^{2} \) |
| 13 | \( 1 - 3.26iT - 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 - 6.25T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 + 8.93iT - 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 - 4.41iT - 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 - 0.818iT - 43T^{2} \) |
| 47 | \( 1 - 3.00iT - 47T^{2} \) |
| 53 | \( 1 + 9.36T + 53T^{2} \) |
| 59 | \( 1 + 4.10T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 9.17iT - 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 5.25iT - 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 + 8.72iT - 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.88166630472741048481511947479, −7.57164791029230279591135937761, −6.56437533001602309380954850936, −6.24422486413411717109610868622, −5.00470973640011110274563348872, −4.48216082169782165037922877318, −3.60115705685989827972155772493, −3.00535363037737795070537534731, −1.86645313126824372120369129156, −0.56080394659100379367164960696,
0.51997999263665695170496959707, 1.78866761818473014018788476403, 3.11833749377308011018463232127, 3.38867043561696802827986516399, 4.50810271963134051412231658370, 5.05582047654570061738640197712, 5.91682763500488835878473827290, 6.93579877354179894505661960011, 7.26590020573801781131535740876, 7.982451397460605940928442770522
