L(s) = 1 | + (−1.70 + 0.301i)3-s + (2.04 + 0.902i)5-s + (1.30 + 2.30i)7-s + (2.81 − 1.02i)9-s + 5.71i·11-s − 3.76·13-s + (−3.76 − 0.922i)15-s − 3.22i·17-s − 0.786i·19-s + (−2.91 − 3.53i)21-s + 1.52·23-s + (3.37 + 3.69i)25-s + (−4.49 + 2.60i)27-s − 6.77i·29-s + 1.56i·31-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.174i)3-s + (0.914 + 0.403i)5-s + (0.492 + 0.870i)7-s + (0.939 − 0.343i)9-s + 1.72i·11-s − 1.04·13-s + (−0.971 − 0.238i)15-s − 0.783i·17-s − 0.180i·19-s + (−0.636 − 0.771i)21-s + 0.318·23-s + (0.674 + 0.738i)25-s + (−0.865 + 0.501i)27-s − 1.25i·29-s + 0.281i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.713910 + 0.942455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.713910 + 0.942455i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.301i)T \) |
| 5 | \( 1 + (-2.04 - 0.902i)T \) |
| 7 | \( 1 + (-1.30 - 2.30i)T \) |
good | 11 | \( 1 - 5.71iT - 11T^{2} \) |
| 13 | \( 1 + 3.76T + 13T^{2} \) |
| 17 | \( 1 + 3.22iT - 17T^{2} \) |
| 19 | \( 1 + 0.786iT - 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 6.77iT - 29T^{2} \) |
| 31 | \( 1 - 1.56iT - 31T^{2} \) |
| 37 | \( 1 - 8.83iT - 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 - 8.85iT - 43T^{2} \) |
| 47 | \( 1 + 1.75iT - 47T^{2} \) |
| 53 | \( 1 - 0.616T + 53T^{2} \) |
| 59 | \( 1 + 6.38T + 59T^{2} \) |
| 61 | \( 1 - 14.4iT - 61T^{2} \) |
| 67 | \( 1 + 6.26iT - 67T^{2} \) |
| 71 | \( 1 - 5.09iT - 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 10.5iT - 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 7.69T + 97T^{2} \) |
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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
−10.17532920249796167077916127263, −9.865783300738871199954698665567, −9.105191749380001654419507007978, −7.62592593329275285964838369762, −6.89529110174459552608214348157, −6.06296781000340126921295335072, −4.97440983334945251558615346507, −4.70693864172483391383890236295, −2.70124724245694961788580114146, −1.72035008609488177618671773006,
0.66309608221197827625960168500, 1.87490254151950470084067857698, 3.61631374074683746790595064364, 4.87025967280074900457831122707, 5.50951276951831454632634521539, 6.35151767419700629761083953459, 7.23945393172908524517156323118, 8.236760698070912298907820546028, 9.173333678897085419946758784078, 10.26053706566223737011946835731