L(s) = 1 | + 2.30i·2-s + (−0.736 − 1.27i)3-s − 3.30·4-s + (−0.733 + 0.423i)5-s + (2.93 − 1.69i)6-s − 3.00i·8-s + (0.414 − 0.718i)9-s + (−0.975 − 1.69i)10-s + (1.30 − 0.751i)11-s + (2.43 + 4.21i)12-s + (2.92 − 2.11i)13-s + (1.08 + 0.624i)15-s + 0.313·16-s + 2.07·17-s + (1.65 + 0.954i)18-s + (−0.0410 − 0.0237i)19-s + ⋯ |
L(s) = 1 | + 1.62i·2-s + (−0.425 − 0.736i)3-s − 1.65·4-s + (−0.328 + 0.189i)5-s + (1.19 − 0.692i)6-s − 1.06i·8-s + (0.138 − 0.239i)9-s + (−0.308 − 0.534i)10-s + (0.392 − 0.226i)11-s + (0.702 + 1.21i)12-s + (0.810 − 0.585i)13-s + (0.279 + 0.161i)15-s + 0.0782·16-s + 0.502·17-s + (0.389 + 0.225i)18-s + (−0.00942 − 0.00544i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.962425 + 0.722943i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.962425 + 0.722943i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.92 + 2.11i)T \) |
good | 2 | \( 1 - 2.30iT - 2T^{2} \) |
| 3 | \( 1 + (0.736 + 1.27i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.733 - 0.423i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.30 + 0.751i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 + (0.0410 + 0.0237i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + (0.679 - 1.17i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.80 - 3.93i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.70iT - 37T^{2} \) |
| 41 | \( 1 + (-8.67 - 5.00i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.311 + 0.180i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.35 - 2.34i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 1.64iT - 59T^{2} \) |
| 61 | \( 1 + (-2.26 + 3.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.76 - 1.02i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.3 + 7.10i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.85 + 3.38i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.5iT - 83T^{2} \) |
| 89 | \( 1 + 17.5iT - 89T^{2} \) |
| 97 | \( 1 + (0.369 - 0.213i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.88280072292119826145758560603, −9.477696570504865534383962501871, −8.729454374580036100239657726726, −7.71587589001812746101230336703, −7.23275839033477671014255157601, −6.28948996969251171522955139376, −5.80358049095509543459584835245, −4.63235155457724008056886878045, −3.33318677175599731727332108813, −0.994786452817822426793597153049,
1.06586463120164338046030277543, 2.50866659681620781884978587218, 3.86983036568855296025897571104, 4.34242314454134630905801923355, 5.39545333067639346326390721609, 6.77605180670282146905929917482, 8.147542259697561622161028094859, 9.142654224641971641421994704748, 9.792927317033908821577208204896, 10.57974345211293859523502005231