L(s) = 1 | + (−0.672 − 1.16i)2-s + (−1.02 + 1.77i)3-s + (0.0951 − 0.164i)4-s + (−1.78 − 3.08i)5-s + 2.75·6-s + (2.62 + 0.349i)7-s − 2.94·8-s + (−0.601 − 1.04i)9-s + (−2.39 + 4.15i)10-s + (−0.639 + 1.10i)11-s + (0.195 + 0.337i)12-s + (−1.35 − 3.29i)14-s + 7.31·15-s + (1.79 + 3.10i)16-s + (−3.86 + 6.70i)17-s + (−0.809 + 1.40i)18-s + ⋯ |
L(s) = 1 | + (−0.475 − 0.823i)2-s + (−0.591 + 1.02i)3-s + (0.0475 − 0.0824i)4-s + (−0.797 − 1.38i)5-s + 1.12·6-s + (0.991 + 0.132i)7-s − 1.04·8-s + (−0.200 − 0.347i)9-s + (−0.758 + 1.31i)10-s + (−0.192 + 0.333i)11-s + (0.0563 + 0.0975i)12-s + (−0.362 − 0.879i)14-s + 1.88·15-s + (0.447 + 0.775i)16-s + (−0.938 + 1.62i)17-s + (−0.190 + 0.330i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7129218781\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7129218781\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.62 - 0.349i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.672 + 1.16i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.02 - 1.77i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.78 + 3.08i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.639 - 1.10i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.86 - 6.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.471 - 0.817i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.823 + 1.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 + (-2.57 + 4.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.528 - 0.914i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.19T + 41T^{2} \) |
| 43 | \( 1 - 3.83T + 43T^{2} \) |
| 47 | \( 1 + (-0.447 - 0.774i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0399 + 0.0692i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.59 - 9.68i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.81 - 6.60i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.16 - 5.47i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + (0.380 - 0.658i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.32T + 83T^{2} \) |
| 89 | \( 1 + (-3.78 - 6.56i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.478T + 97T^{2} \) |
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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
−9.991970443701958862158879237978, −9.057436867972456383790315936055, −8.496722422785398971259938807928, −7.75956279425114542685502132730, −6.14305856942856429796526388781, −5.31784805979867398381896915852, −4.44476896925271428769659529485, −4.02024703669884344244424008220, −2.20458139642227993322079513345, −1.05034315818298038287857736484,
0.46947647501993651453156595848, 2.35870622453074145758492646443, 3.34304062072891275900950187316, 4.76364834268687396113701577559, 5.98018724360060746379171371752, 6.77915621117939573552155712138, 7.19163984250026397953019210788, 7.74558484485730747105210426253, 8.466866005636207118483866406972, 9.557896233962768275746957640873