L(s) = 1 | + (−0.192 − 1.40i)2-s + (−1.92 + 0.540i)4-s − 0.0802i·7-s + (1.12 + 2.59i)8-s + 2.41i·11-s − 5.26·13-s + (−0.112 + 0.0154i)14-s + (3.41 − 2.08i)16-s − 0.255i·17-s − 6.95i·19-s + (3.38 − 0.465i)22-s + 1.64i·23-s + (1.01 + 7.38i)26-s + (0.0433 + 0.154i)28-s + 4.51i·29-s + ⋯ |
L(s) = 1 | + (−0.136 − 0.990i)2-s + (−0.962 + 0.270i)4-s − 0.0303i·7-s + (0.398 + 0.917i)8-s + 0.728i·11-s − 1.46·13-s + (−0.0300 + 0.00413i)14-s + (0.854 − 0.520i)16-s − 0.0620i·17-s − 1.59i·19-s + (0.721 − 0.0993i)22-s + 0.343i·23-s + (0.199 + 1.44i)26-s + (0.00819 + 0.0292i)28-s + 0.838i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.236362244\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.236362244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.192 + 1.40i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.0802iT - 7T^{2} \) |
| 11 | \( 1 - 2.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 + 0.255iT - 17T^{2} \) |
| 19 | \( 1 + 6.95iT - 19T^{2} \) |
| 23 | \( 1 - 1.64iT - 23T^{2} \) |
| 29 | \( 1 - 4.51iT - 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 - 5.70iT - 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 7.27T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 + 9.20T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 8.50iT - 97T^{2} \) |
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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
−9.285052360900927046061185165697, −8.744577556332505209280499671898, −7.52412423731234834212544225992, −7.16042541328084229459642901000, −5.74477281422540390828628909557, −4.71796370176840779318513171816, −4.34130919946326296195264803482, −2.86422576392949610103391477441, −2.35422299584816662131373103370, −0.869133593370219431582328480328,
0.68860630521451244032417688503, 2.38810611935671395645956394562, 3.73343047824367172333356043724, 4.56995337474862477058857821486, 5.51672432372754270995797885207, 6.14207351057883558303509442052, 7.00977054367307533807669580769, 7.897364356863807811610600485424, 8.282096464294146922128286232413, 9.319044558903854163159951057015