L(s) = 1 | − 2.36·2-s + 2.84i·3-s + 3.57·4-s + 5-s − 6.72i·6-s − 0.816i·7-s − 3.71·8-s − 5.11·9-s − 2.36·10-s + (−3.22 + 0.764i)11-s + 10.1i·12-s + 0.847·13-s + 1.92i·14-s + 2.84i·15-s + 1.62·16-s − 4.07i·17-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 1.64i·3-s + 1.78·4-s + 0.447·5-s − 2.74i·6-s − 0.308i·7-s − 1.31·8-s − 1.70·9-s − 0.746·10-s + (−0.973 + 0.230i)11-s + 2.93i·12-s + 0.234·13-s + 0.515i·14-s + 0.735i·15-s + 0.406·16-s − 0.987i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3308279151\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3308279151\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + (3.22 - 0.764i)T \) |
| 19 | \( 1 + (2.17 + 3.77i)T \) |
good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 - 2.84iT - 3T^{2} \) |
| 7 | \( 1 + 0.816iT - 7T^{2} \) |
| 13 | \( 1 - 0.847T + 13T^{2} \) |
| 17 | \( 1 + 4.07iT - 17T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 + 5.91iT - 31T^{2} \) |
| 37 | \( 1 + 9.96iT - 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 5.14iT - 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 1.18iT - 53T^{2} \) |
| 59 | \( 1 - 1.07iT - 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 9.83iT - 67T^{2} \) |
| 71 | \( 1 + 4.42iT - 71T^{2} \) |
| 73 | \( 1 - 4.87iT - 73T^{2} \) |
| 79 | \( 1 - 5.76T + 79T^{2} \) |
| 83 | \( 1 - 11.9iT - 83T^{2} \) |
| 89 | \( 1 + 5.52iT - 89T^{2} \) |
| 97 | \( 1 - 13.9iT - 97T^{2} \) |
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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
−9.490343470330806101433095722754, −9.437609073527839574081806693923, −8.520360248257569357871490046251, −7.58856792013841060294118861789, −6.71194198461996352149205994731, −5.38759174968303517760462080515, −4.67886278444012460391732803836, −3.29683539161338379428162062581, −2.21594827426533810333061694631, −0.26310207880645412734015207024,
1.30433353228533355930398307613, 1.96560823561339406180265264763, 3.05682052469704605457588060665, 5.34299444584273432548571753636, 6.27061798899249473020801919595, 6.93391822676179266514973287235, 7.68355983066743793317003142419, 8.513526603154169276224679320106, 8.717874286373257550106049920252, 10.05509105339275964244567984881