| L(s) = 1 | − i·2-s + 2i·3-s − 4-s + 2·6-s + i·7-s + i·8-s − 9-s − 11-s − 2i·12-s − 2i·13-s + 14-s + 16-s − 6i·17-s + i·18-s − 2·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.15i·3-s − 0.5·4-s + 0.816·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 0.301·11-s − 0.577i·12-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.552985256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.552985256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.016160188086369078198345489357, −7.991962657600809798209103536536, −7.36133733695809529319806056749, −6.12696124048147765798748698660, −5.33912951447419906957897177077, −4.71851625510094114873538830345, −4.03421710480520412941972240603, −3.07883553678661024807136028429, −2.49395615683054007024966929769, −1.00682387342408775416500634629,
0.54841090217872995800557608469, 1.66397717110913711662857335883, 2.60539277417018856901612260098, 3.97978197307482576611617739318, 4.52558211100606653907995824683, 5.67373125888953515517706383635, 6.44394220018345693556867022891, 6.76400685935591825903254017260, 7.55238501891779828730565396992, 8.352863938509922763545728674005
