| L(s) = 1 | + (0.978 + 0.207i)2-s + (−1.02 − 1.77i)3-s + (0.913 + 0.406i)4-s + (0.131 + 1.24i)5-s + (−0.633 − 1.94i)6-s + (−2.58 − 0.563i)7-s + (0.809 + 0.587i)8-s + (−0.598 + 1.03i)9-s + (−0.131 + 1.24i)10-s + (0.512 − 4.87i)11-s + (−0.214 − 2.03i)12-s + (−0.169 − 0.522i)13-s + (−2.41 − 1.08i)14-s + (2.07 − 1.51i)15-s + (0.669 + 0.743i)16-s + (0.727 − 6.92i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (−0.591 − 1.02i)3-s + (0.456 + 0.203i)4-s + (0.0586 + 0.557i)5-s + (−0.258 − 0.795i)6-s + (−0.977 − 0.213i)7-s + (0.286 + 0.207i)8-s + (−0.199 + 0.345i)9-s + (−0.0414 + 0.394i)10-s + (0.154 − 1.46i)11-s + (−0.0618 − 0.588i)12-s + (−0.0470 − 0.144i)13-s + (−0.644 − 0.290i)14-s + (0.536 − 0.390i)15-s + (0.167 + 0.185i)16-s + (0.176 − 1.67i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.875991 - 1.13244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.875991 - 1.13244i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 7 | \( 1 + (2.58 + 0.563i)T \) |
| 41 | \( 1 + (3.15 - 5.57i)T \) |
| good | 3 | \( 1 + (1.02 + 1.77i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.131 - 1.24i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.512 + 4.87i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.169 + 0.522i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.727 + 6.92i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (1.45 + 1.61i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (1.68 + 0.358i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-1.13 + 0.822i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.162 - 1.54i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.563 - 5.36i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (1.79 + 5.52i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (5.48 + 1.16i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-8.24 - 3.67i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-4.04 + 4.49i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.24 - 1.38i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-6.26 - 2.78i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-8.98 - 6.52i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (7.30 + 12.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.67 + 8.09i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.26T + 83T^{2} \) |
| 89 | \( 1 + (-8.02 - 8.91i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (7.75 - 5.63i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.76481711678450234936013973618, −9.736831175306372811531589111467, −8.520713310063683670131298035741, −7.32583348919142885824728011804, −6.66326254250399977262857359222, −6.16505153263039892308841041827, −5.12084999611727982279095233570, −3.52727567188696546612247596432, −2.68957726756044802486309732836, −0.69024412548255204543053857151,
1.98219345626260806823794429600, 3.69706313214512865865120665409, 4.35892200728682957696566221221, 5.29864941043201387296904198186, 6.13189648944138066643351222759, 7.12287294290526127659932318688, 8.509758920353025102341430655095, 9.665918144567334109594311950312, 10.09172315248016862685902471715, 10.88049731572802899508726649126
