| L(s) = 1 | − i·2-s − 4-s + 0.469·5-s + (1.12 − 2.39i)7-s + i·8-s − 0.469i·10-s − 0.559i·11-s − 1.49i·13-s + (−2.39 − 1.12i)14-s + 16-s − 5.16·17-s + 4.33i·19-s − 0.469·20-s − 0.559·22-s − 0.835i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.209·5-s + (0.426 − 0.904i)7-s + 0.353i·8-s − 0.148i·10-s − 0.168i·11-s − 0.413i·13-s + (−0.639 − 0.301i)14-s + 0.250·16-s − 1.25·17-s + 0.995i·19-s − 0.104·20-s − 0.119·22-s − 0.174i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2033019944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2033019944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.12 + 2.39i)T \) |
| 29 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.469T + 5T^{2} \) |
| 11 | \( 1 + 0.559iT - 11T^{2} \) |
| 13 | \( 1 + 1.49iT - 13T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 - 4.33iT - 19T^{2} \) |
| 23 | \( 1 + 0.835iT - 23T^{2} \) |
| 31 | \( 1 - 2.13iT - 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 - 8.11T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 - 8.29iT - 53T^{2} \) |
| 59 | \( 1 - 4.55T + 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 7.58iT - 71T^{2} \) |
| 73 | \( 1 + 4.35iT - 73T^{2} \) |
| 79 | \( 1 + 6.48T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 16.8iT - 97T^{2} \) |
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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
−8.235085788247442389258295231067, −7.37602907501511787775179088150, −6.59181083462826943173545900288, −5.66426829346148724251099791703, −4.84462659931399032335123859709, −4.04607320316763325943957444397, −3.38718494974352644841476758182, −2.20181995905522504513779478548, −1.38970490492091829534179281925, −0.05722493038269146864662752951,
1.73289787592504164931986608758, 2.55266811618124654634575840981, 3.79671406128177026858212904224, 4.71412499799969246204708870134, 5.28473095816450317292339460236, 6.10300377408613154496519391322, 6.81535973119179570821027336894, 7.42835246767538065266008131988, 8.547122972426635005734194134656, 8.716496038744410581553346617352
