L(s) = 1 | + (−1.63 + 0.573i)3-s + (−0.707 − 0.707i)7-s + (2.34 − 1.87i)9-s − 6.10i·11-s + (−2.13 + 2.13i)13-s + (1.21 − 1.21i)17-s − 0.390i·19-s + (1.56 + 0.750i)21-s + (2.04 + 2.04i)23-s + (−2.75 + 4.40i)27-s − 5.67·29-s + 1.37·31-s + (3.50 + 9.97i)33-s + (−2.84 − 2.84i)37-s + (2.26 − 4.71i)39-s + ⋯ |
L(s) = 1 | + (−0.943 + 0.331i)3-s + (−0.267 − 0.267i)7-s + (0.780 − 0.624i)9-s − 1.84i·11-s + (−0.591 + 0.591i)13-s + (0.294 − 0.294i)17-s − 0.0895i·19-s + (0.340 + 0.163i)21-s + (0.426 + 0.426i)23-s + (−0.529 + 0.848i)27-s − 1.05·29-s + 0.246·31-s + (0.609 + 1.73i)33-s + (−0.467 − 0.467i)37-s + (0.362 − 0.754i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1294841128\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1294841128\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.63 - 0.573i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + 6.10iT - 11T^{2} \) |
| 13 | \( 1 + (2.13 - 2.13i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.21 + 1.21i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.390iT - 19T^{2} \) |
| 23 | \( 1 + (-2.04 - 2.04i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 - 1.37T + 31T^{2} \) |
| 37 | \( 1 + (2.84 + 2.84i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.59iT - 41T^{2} \) |
| 43 | \( 1 + (3.04 - 3.04i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.32 + 1.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.10 - 9.10i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 + (7.63 + 7.63i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.26iT - 71T^{2} \) |
| 73 | \( 1 + (11.3 - 11.3i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.56iT - 79T^{2} \) |
| 83 | \( 1 + (3.59 + 3.59i)T + 83iT^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + (10.3 + 10.3i)T + 97iT^{2} \) |
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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
−8.991155149005848697799943590131, −7.81472698130055814765181812370, −7.06025400315520793260833006501, −6.18475197181370425220658503420, −5.63147722774275808965218353129, −4.75486183746527583884198357541, −3.79941665058031900288114996049, −2.97032247124866712511898048105, −1.27526509594012382827022655189, −0.05527878687583831089874775777,
1.57018279966953938016397143560, 2.51680409979633181636023398637, 3.94565057400337361397757561390, 4.88006166775993587710740474823, 5.44296897362393510288808406588, 6.39836016396056629203873173723, 7.21188718251409562820430924140, 7.58680949426346761844640758975, 8.737808259177562972697300003081, 9.755169959396363738939663949231