| L(s) = 1 | − 3.32i·3-s − i·5-s + 7-s − 8.06·9-s + 3.78i·11-s − 4.49i·13-s − 3.32·15-s − 2.58·17-s − 5.09i·19-s − 3.32i·21-s − 2.16·23-s − 25-s + 16.8i·27-s + 0.164i·29-s − 6.36·31-s + ⋯ |
| L(s) = 1 | − 1.92i·3-s − 0.447i·5-s + 0.377·7-s − 2.68·9-s + 1.14i·11-s − 1.24i·13-s − 0.858·15-s − 0.628·17-s − 1.16i·19-s − 0.725i·21-s − 0.450·23-s − 0.200·25-s + 3.24i·27-s + 0.0305i·29-s − 1.14·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5939229378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5939229378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 3.32iT - 3T^{2} \) |
| 11 | \( 1 - 3.78iT - 11T^{2} \) |
| 13 | \( 1 + 4.49iT - 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 5.09iT - 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 0.164iT - 29T^{2} \) |
| 31 | \( 1 + 6.36T + 31T^{2} \) |
| 37 | \( 1 - 7.95iT - 37T^{2} \) |
| 41 | \( 1 - 3.23T + 41T^{2} \) |
| 43 | \( 1 - 8.69iT - 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 6.26iT - 53T^{2} \) |
| 59 | \( 1 + 6.88iT - 59T^{2} \) |
| 61 | \( 1 + 8.08iT - 61T^{2} \) |
| 67 | \( 1 + 2.05iT - 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 7.80T + 73T^{2} \) |
| 79 | \( 1 + 6.08T + 79T^{2} \) |
| 83 | \( 1 - 17.7iT - 83T^{2} \) |
| 89 | \( 1 - 4.16T + 89T^{2} \) |
| 97 | \( 1 + 0.338T + 97T^{2} \) |
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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
−8.120683191247293914348439655933, −7.82827065290353810589745804898, −6.94141098168141425428861847671, −6.40961993111988343429034381310, −5.38703911022075766416202226084, −4.72269445342661720342579323937, −3.14040685404480198961378369755, −2.20140693481831132865741670529, −1.40540472948436356812306330320, −0.19640173202547372737149073819,
2.11489702020860512384701929009, 3.31680552289003708218022117260, 3.93628054989872179300389008961, 4.58131588056083321981912670374, 5.67106432433128889157544073077, 6.05361052429895377962766731274, 7.33363864571509670130206086731, 8.426936279557704440716581607193, 8.927418278486534676363426125355, 9.537417636715297380876754121605
