L(s) = 1 | + (−0.5 + 0.866i)5-s + (1.62 + 2.09i)7-s + (−2.12 − 3.67i)11-s + 3.24·13-s + (2.12 + 3.67i)17-s + (3.5 − 6.06i)19-s + (−2.12 + 3.67i)23-s + (−0.499 − 0.866i)25-s + 1.75·29-s + (4.74 + 8.21i)31-s + (−2.62 + 0.358i)35-s + (−1.62 + 2.80i)37-s + 4.24·41-s + 3.24·43-s + (−3 + 5.19i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.612 + 0.790i)7-s + (−0.639 − 1.10i)11-s + 0.899·13-s + (0.514 + 0.891i)17-s + (0.802 − 1.39i)19-s + (−0.442 + 0.766i)23-s + (−0.0999 − 0.173i)25-s + 0.326·29-s + (0.851 + 1.47i)31-s + (−0.443 + 0.0606i)35-s + (−0.266 + 0.461i)37-s + 0.662·41-s + 0.494·43-s + (−0.437 + 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.712818290\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.712818290\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.62 - 2.09i)T \) |
good | 11 | \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 17 | \( 1 + (-2.12 - 3.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.12 - 3.67i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 + (-4.74 - 8.21i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.62 - 2.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.24 - 7.34i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.12 + 8.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.24 - 3.88i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.62 - 4.54i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + (4.62 + 8.00i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + (5.12 - 8.87i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.485T + 97T^{2} \) |
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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
−9.770247853907568020374926589454, −8.740340814384956135065512696791, −8.290521503060328428408954730525, −7.46664122250502789298287201201, −6.31123394377091446421841536434, −5.65521055207275930482434541682, −4.77369789118127068714612529079, −3.45965757342017302028359926700, −2.72897962206069743905976128045, −1.22003362621220431944909068351,
0.878529131067263648826946604968, 2.15007980847801511799241417155, 3.61402089681465346180268872021, 4.43972873644595143326666936391, 5.24305907986460708846884144149, 6.24575771342782061880304356322, 7.48111374356440582283825032620, 7.77516537911387945102871271155, 8.659800891764323070644507428374, 9.844344911162434351305636934568