L(s) = 1 | + (0.707 + 0.707i)2-s + (0.541 + 0.541i)3-s + 1.00i·4-s + (−0.382 + 0.923i)5-s + 0.765i·6-s + (−0.707 + 0.707i)8-s − 0.414i·9-s + (−0.923 + 0.382i)10-s + (−0.541 + 0.541i)12-s + (1.30 + 1.30i)13-s + (−0.707 + 0.292i)15-s − 1.00·16-s + (0.292 − 0.292i)18-s − 1.84·19-s + (−0.923 − 0.382i)20-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.541 + 0.541i)3-s + 1.00i·4-s + (−0.382 + 0.923i)5-s + 0.765i·6-s + (−0.707 + 0.707i)8-s − 0.414i·9-s + (−0.923 + 0.382i)10-s + (−0.541 + 0.541i)12-s + (1.30 + 1.30i)13-s + (−0.707 + 0.292i)15-s − 1.00·16-s + (0.292 − 0.292i)18-s − 1.84·19-s + (−0.923 − 0.382i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.819452162\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.819452162\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + 1.84T + T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 - 0.765iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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
−9.427833427905069703318939483093, −8.595579878357784901600956938813, −8.339635515367780932620040151922, −6.95214609369200382555413695105, −6.64483578941099993140807543395, −5.93132612766423035403059356527, −4.43769194837210228365651602727, −4.07298476243718887714131304350, −3.25816020492559746620396217391, −2.30548920466369016886438164751,
1.05957234323696511752267952411, 2.04559702662300362621936539943, 3.17865087893250608369777441658, 3.96178447478141464959099891846, 4.94626264280185424526874393091, 5.62989209881800380371332003825, 6.55686269860475933621454450747, 7.64144834463323418668963833641, 8.452254159979453597341868839412, 8.848246210970120394711408458302