| L(s) = 1 | + (−1.43 + 2.48i)5-s + (2.56 + 0.632i)7-s + (2.34 − 1.35i)11-s + (−3.18 + 1.84i)13-s + (−3.22 + 5.58i)17-s + (2.73 − 1.58i)19-s + (−2.59 − 1.49i)23-s + (−1.61 − 2.79i)25-s + (2.48 + 1.43i)29-s + 9.54i·31-s + (−5.25 + 5.47i)35-s + (−1.70 − 2.95i)37-s + (0.794 + 1.37i)41-s + (−4.67 + 8.10i)43-s − 11.3·47-s + ⋯ |
| L(s) = 1 | + (−0.641 + 1.11i)5-s + (0.970 + 0.239i)7-s + (0.708 − 0.408i)11-s + (−0.884 + 0.510i)13-s + (−0.781 + 1.35i)17-s + (0.628 − 0.362i)19-s + (−0.540 − 0.311i)23-s + (−0.322 − 0.558i)25-s + (0.461 + 0.266i)29-s + 1.71i·31-s + (−0.888 + 0.925i)35-s + (−0.280 − 0.485i)37-s + (0.124 + 0.214i)41-s + (−0.713 + 1.23i)43-s − 1.64·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.822889 + 0.978818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.822889 + 0.978818i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.56 - 0.632i)T \) |
| good | 5 | \( 1 + (1.43 - 2.48i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.34 + 1.35i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.18 - 1.84i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.73 + 1.58i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 + 1.49i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.48 - 1.43i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 9.54iT - 31T^{2} \) |
| 37 | \( 1 + (1.70 + 2.95i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.794 - 1.37i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.67 - 8.10i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + (2.16 + 1.24i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 8.67T + 59T^{2} \) |
| 61 | \( 1 - 0.654iT - 61T^{2} \) |
| 67 | \( 1 - 7.72T + 67T^{2} \) |
| 71 | \( 1 - 7.86iT - 71T^{2} \) |
| 73 | \( 1 + (-11.0 - 6.39i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 + (-7.92 + 13.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.14 + 5.45i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.2 + 7.62i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.78331201526381919807482326102, −9.834340371153230767179378156741, −8.683517720805484127668884962517, −8.064183144850549051145027074750, −6.99498847064877453627254927849, −6.46377702938298272919641624920, −5.09585418053180008543976978407, −4.12447279911787532786444593018, −3.06922329592239855497029614055, −1.76668405726467895175884972035,
0.66877481718602134977029341248, 2.13384051958293600844697606838, 3.82641319217453955309114888052, 4.75191690127835785384301937048, 5.24757670294855985358073889790, 6.77450130064263603311358978743, 7.73805522023253603926942197584, 8.231364189379100692569069206315, 9.294181951737309405740098251986, 9.898880346843021782985064201589
