| L(s) = 1 | + 1.85i·3-s − 5-s + (−1.29 − 2.30i)7-s − 0.432·9-s + 2.01·11-s − 1.82·13-s − 1.85i·15-s + 1.64i·17-s + 1.26i·19-s + (4.27 − 2.40i)21-s − 3.19i·23-s + 25-s + 4.75i·27-s + 4.57i·29-s + 0.834·31-s + ⋯ |
| L(s) = 1 | + 1.06i·3-s − 0.447·5-s + (−0.490 − 0.871i)7-s − 0.144·9-s + 0.608·11-s − 0.507·13-s − 0.478i·15-s + 0.399i·17-s + 0.289i·19-s + (0.931 − 0.525i)21-s − 0.665i·23-s + 0.200·25-s + 0.915i·27-s + 0.849i·29-s + 0.149·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.074970156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074970156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (1.29 + 2.30i)T \) |
| good | 3 | \( 1 - 1.85iT - 3T^{2} \) |
| 11 | \( 1 - 2.01T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 1.64iT - 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 + 3.19iT - 23T^{2} \) |
| 29 | \( 1 - 4.57iT - 29T^{2} \) |
| 31 | \( 1 - 0.834T + 31T^{2} \) |
| 37 | \( 1 - 7.82iT - 37T^{2} \) |
| 41 | \( 1 - 2.75iT - 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 8.65iT - 53T^{2} \) |
| 59 | \( 1 + 3.88iT - 59T^{2} \) |
| 61 | \( 1 - 0.285T + 61T^{2} \) |
| 67 | \( 1 + 3.94T + 67T^{2} \) |
| 71 | \( 1 + 11.1iT - 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 7.74iT - 83T^{2} \) |
| 89 | \( 1 - 8.39iT - 89T^{2} \) |
| 97 | \( 1 + 9.32iT - 97T^{2} \) |
| 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
−9.537720619862217659716503837908, −8.661810786114322575392469084992, −7.81168845133127612736257073427, −6.95268416969458075372717462867, −6.31559779704533887721470822553, −5.06859071690088443826323224388, −4.37346776379923872040484005311, −3.77320879049634875895630694311, −2.95558330517586103023142745562, −1.25990420208060342697790774660,
0.39833459160206649891378801087, 1.78718877567252109442195324420, 2.66386535942987846427087934060, 3.72165233665772374352447628956, 4.78507497401834452250308567938, 5.81770839857802443481200793708, 6.48810289648041335262510403788, 7.25905274310591834116923542688, 7.78073730363620855037734420014, 8.726947507865307419367861649198
