| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s − 4.63i·7-s + i·8-s − 9-s − 2.53·11-s − i·12-s − 0.143i·13-s − 4.63·14-s + 16-s − 7.49i·17-s + i·18-s + 6.74·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.75i·7-s + 0.353i·8-s − 0.333·9-s − 0.765·11-s − 0.288i·12-s − 0.0399i·13-s − 1.23·14-s + 0.250·16-s − 1.81i·17-s + 0.235i·18-s + 1.54·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.014184278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014184278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.63iT - 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 + 0.143iT - 13T^{2} \) |
| 17 | \( 1 + 7.49iT - 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 - 1.67iT - 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 - 0.0889iT - 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 9.02iT - 43T^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 + 4.96iT - 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 7.64iT - 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 1.96iT - 73T^{2} \) |
| 79 | \( 1 + 0.747T + 79T^{2} \) |
| 83 | \( 1 + 3.10iT - 83T^{2} \) |
| 89 | \( 1 + 0.733T + 89T^{2} \) |
| 97 | \( 1 - 9.12iT - 97T^{2} \) |
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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
−8.065993936748929092996370436069, −7.48974928811878898826640595448, −6.89573930269816132643998446753, −5.58428330759117214097866619615, −4.86412056756098613442862993838, −4.31375159959232663961517448528, −3.30538916657595053414936703713, −2.83213482483229004288224019125, −1.26412028825572977392149177959, −0.32008783593779358946772505901,
1.44926520594701462106325654162, 2.50944303126813358593484822218, 3.29387538869328086468567633286, 4.59040882792947409502270734361, 5.45279471515296994443349861333, 5.92963874598013214104414867956, 6.47222184225668420339953860403, 7.65068172146902008054406023819, 7.952231130076924793872959538246, 8.831120133757749446718011305861
