L(s) = 1 | + (1.87 + 1.87i)3-s + (2.44 + 2.44i)7-s + 4i·9-s − 4.58·11-s + (3.74 − 3.74i)13-s + (−1.22 + 1.22i)17-s + 4.58i·19-s + 9.16i·21-s + (−4.89 + 4.89i)23-s + (−1.87 + 1.87i)27-s + 9.16·29-s + 4i·31-s + (−8.57 − 8.57i)33-s + (3.74 + 3.74i)37-s + 14·39-s + ⋯ |
L(s) = 1 | + (1.08 + 1.08i)3-s + (0.925 + 0.925i)7-s + 1.33i·9-s − 1.38·11-s + (1.03 − 1.03i)13-s + (−0.297 + 0.297i)17-s + 1.05i·19-s + 1.99i·21-s + (−1.02 + 1.02i)23-s + (−0.360 + 0.360i)27-s + 1.70·29-s + 0.718i·31-s + (−1.49 − 1.49i)33-s + (0.615 + 0.615i)37-s + 2.24·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.530139339\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.530139339\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.87 - 1.87i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 + (-3.74 + 3.74i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.22 - 1.22i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.58iT - 19T^{2} \) |
| 23 | \( 1 + (4.89 - 4.89i)T - 23iT^{2} \) |
| 29 | \( 1 - 9.16T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + (-3.74 - 3.74i)T + 37iT^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + (-3.74 - 3.74i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.44 + 2.44i)T + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 9.16iT - 61T^{2} \) |
| 67 | \( 1 + (-1.87 + 1.87i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (6.12 + 6.12i)T + 73iT^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + (5.61 + 5.61i)T + 83iT^{2} \) |
| 89 | \( 1 - 15iT - 89T^{2} \) |
| 97 | \( 1 + (4.89 - 4.89i)T - 97iT^{2} \) |
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
−9.705385682370727374359184230759, −8.602372965554057585332282595680, −8.206585549812395641022374988563, −7.88785349878126012894433005725, −6.17666699110841186185091909223, −5.34029441989222616319714471477, −4.68318921441990807150409978911, −3.54043018772672724910451368093, −2.87047782001052207087386404328, −1.79147956377085172314531382710,
0.866781258184841628813657139805, 2.04617060391975661938679765567, 2.76489029345385201306016995294, 4.08528543690129684375526051225, 4.81882454870126695317065301699, 6.20487660378188230645278795576, 7.03236899654495641497823893351, 7.59785301446287337965227026755, 8.448773378923763358097522342142, 8.651843136060175171922247298824