L(s) = 1 | + (0.0511 + 0.0886i)5-s + (−1.64 + 2.85i)7-s + (0.5 − 0.866i)11-s + (1.31 + 2.27i)13-s − 4.21·17-s + 4.73·19-s + (1.31 + 2.28i)23-s + (2.49 − 4.32i)25-s + (4.68 − 8.12i)29-s + (2.66 + 4.61i)31-s − 0.337·35-s − 4.31·37-s + (4.53 + 7.85i)41-s + (0.811 − 1.40i)43-s + (−4.76 + 8.24i)47-s + ⋯ |
L(s) = 1 | + (0.0228 + 0.0396i)5-s + (−0.622 + 1.07i)7-s + (0.150 − 0.261i)11-s + (0.364 + 0.631i)13-s − 1.02·17-s + 1.08·19-s + (0.274 + 0.475i)23-s + (0.498 − 0.864i)25-s + (0.870 − 1.50i)29-s + (0.478 + 0.829i)31-s − 0.0569·35-s − 0.709·37-s + (0.708 + 1.22i)41-s + (0.123 − 0.214i)43-s + (−0.694 + 1.20i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.327683455\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.327683455\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.0511 - 0.0886i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.64 - 2.85i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-1.31 - 2.27i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.21T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + (-1.31 - 2.28i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.68 + 8.12i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.66 - 4.61i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 + (-4.53 - 7.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.811 + 1.40i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.76 - 8.24i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.801T + 53T^{2} \) |
| 59 | \( 1 + (2.50 + 4.34i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.60 - 13.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.21 - 3.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 7.68T + 73T^{2} \) |
| 79 | \( 1 + (4.81 - 8.34i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.80 + 10.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 + (8.72 - 15.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.873014213754582998494668253964, −8.199686993218114532261102279262, −7.24103953916178071285561265175, −6.30691475678756823796553770473, −6.08514804015057941595725405260, −4.96553976575623332602768377491, −4.24191849227784297266885870003, −3.07351444578470008432912155507, −2.52781175565576002864434211355, −1.22878475269251765881270728428,
0.42765179982934110536707006898, 1.52190324896956344555048037995, 2.92986495829547170894015548309, 3.55783368253166483244459548105, 4.49000377762263739585735274843, 5.23746755244075563074657624942, 6.21977622445127278800561312424, 7.00678261783623368045651257726, 7.35392687331691100216346157899, 8.395701363492292279388737575649