| L(s) = 1 | + i·2-s − i·3-s − 4-s − 2.23·5-s + 6-s + 4i·7-s − i·8-s − 9-s − 2.23i·10-s + i·12-s − 6.47i·13-s − 4·14-s + 2.23i·15-s + 16-s − 6.47i·17-s − i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.999·5-s + 0.408·6-s + 1.51i·7-s − 0.353i·8-s − 0.333·9-s − 0.707i·10-s + 0.288i·12-s − 1.79i·13-s − 1.06·14-s + 0.577i·15-s + 0.250·16-s − 1.56i·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.380926 - 0.380926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.380926 - 0.380926i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + 2.23T \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.47iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8.47iT - 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 12.9iT - 47T^{2} \) |
| 53 | \( 1 + 8.94iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 12.9iT - 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 - 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 3.52T + 79T^{2} \) |
| 83 | \( 1 + 10.4iT - 83T^{2} \) |
| 89 | \( 1 - 7.52T + 89T^{2} \) |
| 97 | \( 1 - 13.4iT - 97T^{2} \) |
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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
−10.14192352777288254623645995821, −8.970949381782558784857983572564, −8.435164371446865427441259973072, −7.60043853056113788123713051930, −6.92081423521753066300261229420, −5.55813694762491457752312204268, −5.26615454674022250018664773421, −3.59843096802749734320889221387, −2.51464481820494962851350412277, −0.28927185579166921308894038430,
1.57935004160141494179517909469, 3.43413035034542728602701926006, 4.15995926498101660216621074937, 4.57427548409362316011750650819, 6.32153899500014969128353281619, 7.29352978791800439323725143738, 8.225097180190008683413781403614, 9.080257833662598238013344025118, 10.04319459277055068559300047233, 10.81828253308088208587845123004
