L(s) = 1 | + (0.647 − 1.99i)2-s + (1.54 − 1.12i)3-s + (−1.93 − 1.40i)4-s + (−1.23 − 3.81i)6-s + (−2.48 − 1.80i)7-s + (−0.661 + 0.480i)8-s + (0.203 − 0.627i)9-s + (1.86 + 2.74i)11-s − 4.57·12-s + (−0.942 + 2.90i)13-s + (−5.19 + 3.77i)14-s + (−0.947 − 2.91i)16-s + (0.143 + 0.441i)17-s + (−1.11 − 0.812i)18-s + (6.38 − 4.64i)19-s + ⋯ |
L(s) = 1 | + (0.457 − 1.40i)2-s + (0.893 − 0.649i)3-s + (−0.966 − 0.702i)4-s + (−0.505 − 1.55i)6-s + (−0.937 − 0.681i)7-s + (−0.233 + 0.169i)8-s + (0.0679 − 0.209i)9-s + (0.561 + 0.827i)11-s − 1.31·12-s + (−0.261 + 0.804i)13-s + (−1.38 + 1.00i)14-s + (−0.236 − 0.729i)16-s + (0.0347 + 0.106i)17-s + (−0.263 − 0.191i)18-s + (1.46 − 1.06i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.611954 - 1.83938i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.611954 - 1.83938i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + (-1.86 - 2.74i)T \) |
good | 2 | \( 1 + (-0.647 + 1.99i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.54 + 1.12i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (2.48 + 1.80i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (0.942 - 2.90i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.143 - 0.441i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-6.38 + 4.64i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 + (3.01 + 2.18i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (3.23 - 9.96i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.49 - 1.08i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.56 + 2.59i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 1.31T + 43T^{2} \) |
| 47 | \( 1 + (2.41 - 1.75i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (1.29 - 3.98i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (2.27 + 1.65i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.623 + 1.91i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 6.75T + 67T^{2} \) |
| 71 | \( 1 + (2.01 + 6.20i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (7.98 + 5.80i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.57 - 10.9i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (2.75 + 8.48i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 + (4.74 - 14.6i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.68704013561879202867696199157, −10.70131665591185566438274422995, −9.616718353027791736067253669353, −9.126396807783093875379310599762, −7.40822538844591646731611789659, −6.89395756841047728752306049638, −4.86731643464404126852555282794, −3.66141903400689684166408717843, −2.73981401483663508296916555633, −1.46373662610282744122650657398,
3.03118753377796157755362928910, 3.93957424927048787794875353354, 5.52427607170676065108059704069, 6.09319748287351302608329292518, 7.41919546738569644473627101320, 8.308121868928898908856799457357, 9.223676011226257414247149403156, 9.879794232603767707038277517870, 11.39840518685438167127122964631, 12.64899657281384858641745418382