| L(s) = 1 | + (−1.65 − 0.5i)3-s − 3.31i·5-s + (2.5 + 1.65i)9-s − 3.31·11-s + 4·13-s + (−1.65 + 5.5i)15-s + 3.31i·17-s − 7i·19-s − 3.31·23-s − 6·25-s + (−3.31 − 4i)27-s − 6.63i·29-s − 3i·31-s + (5.5 + 1.65i)33-s − 37-s + ⋯ |
| L(s) = 1 | + (−0.957 − 0.288i)3-s − 1.48i·5-s + (0.833 + 0.552i)9-s − 1.00·11-s + 1.10·13-s + (−0.428 + 1.42i)15-s + 0.804i·17-s − 1.60i·19-s − 0.691·23-s − 1.20·25-s + (−0.638 − 0.769i)27-s − 1.23i·29-s − 0.538i·31-s + (0.957 + 0.288i)33-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5793254181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5793254181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.65 + 0.5i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.31iT - 5T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 3.31iT - 17T^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 + 6.63iT - 29T^{2} \) |
| 31 | \( 1 + 3iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 6.63iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 9.94T + 47T^{2} \) |
| 53 | \( 1 - 3.31iT - 53T^{2} \) |
| 59 | \( 1 + 3.31T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 9iT - 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 3.31iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.532028768768295493799085800220, −7.87161218401404471821777313127, −7.00959279316253872504587968709, −5.84259545461250534936219790857, −5.62983841764269168220642875609, −4.57934455026507353882062436082, −4.07675278987242159295344336847, −2.37320654341794681629595739375, −1.22256879844287843679056171104, −0.24510044715582581986988388617,
1.55834919507136475650581347254, 2.96126512595993418245161730918, 3.62892121347177518730254026326, 4.68696232001401590122359276730, 5.77399626508176851761961432814, 6.11538298672732416983799848957, 7.05900978394595962758690509627, 7.60892742034007562579940813487, 8.599650862724532262779366082653, 9.734381130549487021282100097473
