| L(s) = 1 | + 4i·7-s + 3·9-s − 4·11-s + 2i·13-s + 2i·17-s + 4·19-s + 4i·23-s + 2·29-s + 8·31-s + 6i·37-s − 6·41-s − 8i·43-s − 4i·47-s − 9·49-s − 6i·53-s + ⋯ |
| L(s) = 1 | + 1.51i·7-s + 9-s − 1.20·11-s + 0.554i·13-s + 0.485i·17-s + 0.917·19-s + 0.834i·23-s + 0.371·29-s + 1.43·31-s + 0.986i·37-s − 0.937·41-s − 1.21i·43-s − 0.583i·47-s − 1.28·49-s − 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12940 + 0.698012i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12940 + 0.698012i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−11.66637468486795400945844347698, −10.36371787716092953955720282987, −9.687941945913011033985574344810, −8.669159741380960439502819043824, −7.82345485548024244995230582501, −6.69750607834347011257043200244, −5.57782202616065436043968572110, −4.74860444575389548886244759317, −3.16295329624880242409614353354, −1.89590827212659631097713103959,
0.939278969634963650164151672092, 2.89662801195393895582887534117, 4.22963467981581805606017116424, 5.08020078559631303189439793818, 6.56463226477988062593447218137, 7.49266168316699170013362263022, 8.019356680266294677456073115161, 9.569900651218649872082119047581, 10.34432295746027101544622215853, 10.77212921437032853485016691296
