L(s) = 1 | + (−1.26 − 1.26i)3-s + (−2.95 + 2.95i)5-s + i·7-s + 0.189i·9-s + (−3.18 + 3.18i)11-s + (−3.42 − 3.42i)13-s + 7.46·15-s − 5.13·17-s + (−1.50 − 1.50i)19-s + (1.26 − 1.26i)21-s + 7.11i·23-s − 12.4i·25-s + (−3.54 + 3.54i)27-s + (3.84 + 3.84i)29-s + 0.831·31-s + ⋯ |
L(s) = 1 | + (−0.729 − 0.729i)3-s + (−1.32 + 1.32i)5-s + 0.377i·7-s + 0.0630i·9-s + (−0.960 + 0.960i)11-s + (−0.950 − 0.950i)13-s + 1.92·15-s − 1.24·17-s + (−0.345 − 0.345i)19-s + (0.275 − 0.275i)21-s + 1.48i·23-s − 2.49i·25-s + (−0.683 + 0.683i)27-s + (0.714 + 0.714i)29-s + 0.149·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3688138770\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3688138770\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (1.26 + 1.26i)T + 3iT^{2} \) |
| 5 | \( 1 + (2.95 - 2.95i)T - 5iT^{2} \) |
| 11 | \( 1 + (3.18 - 3.18i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.42 + 3.42i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.13T + 17T^{2} \) |
| 19 | \( 1 + (1.50 + 1.50i)T + 19iT^{2} \) |
| 23 | \( 1 - 7.11iT - 23T^{2} \) |
| 29 | \( 1 + (-3.84 - 3.84i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.831T + 31T^{2} \) |
| 37 | \( 1 + (-5.64 + 5.64i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.22iT - 41T^{2} \) |
| 43 | \( 1 + (-1.61 + 1.61i)T - 43iT^{2} \) |
| 47 | \( 1 - 7.83T + 47T^{2} \) |
| 53 | \( 1 + (5.58 - 5.58i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.85 + 1.85i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.65 - 1.65i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.77 - 5.77i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.04iT - 71T^{2} \) |
| 73 | \( 1 + 7.67iT - 73T^{2} \) |
| 79 | \( 1 + 1.90T + 79T^{2} \) |
| 83 | \( 1 + (7.97 + 7.97i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.49iT - 89T^{2} \) |
| 97 | \( 1 + 1.98T + 97T^{2} \) |
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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
−9.154046488892680280604767010261, −7.966948617618332448237585193804, −7.36371993266052248932847947625, −7.04949522574098839840244705023, −6.12514759377750065218023891829, −5.16304277271855522962922984132, −4.18865151045922188943283942543, −3.02442230837739381050481208664, −2.27320440729567156631384211043, −0.27064448697700262893930889310,
0.61043625365112520664514292815, 2.49026728781164723607037989771, 4.00604475140191050218705043786, 4.57351428745285046917969720808, 4.91509557371277138312003566690, 6.03840187992361118754204808730, 7.06339120671601048885349728795, 8.189679247926794829818945230632, 8.343810389352857587238739162450, 9.399965626312631540529619127581