L(s) = 1 | − 2-s + (−0.707 − 0.707i)3-s + 4-s + (−0.530 − 2.17i)5-s + (0.707 + 0.707i)6-s + (−1.93 − 1.93i)7-s − 8-s + 1.00i·9-s + (0.530 + 2.17i)10-s + 2.54i·11-s + (−0.707 − 0.707i)12-s − 2.64·13-s + (1.93 + 1.93i)14-s + (−1.16 + 1.91i)15-s + 16-s + 3.60i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (−0.237 − 0.971i)5-s + (0.288 + 0.288i)6-s + (−0.730 − 0.730i)7-s − 0.353·8-s + 0.333i·9-s + (0.167 + 0.686i)10-s + 0.766i·11-s + (−0.204 − 0.204i)12-s − 0.732·13-s + (0.516 + 0.516i)14-s + (−0.299 + 0.493i)15-s + 0.250·16-s + 0.874i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3987078212\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3987078212\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.530 + 2.17i)T \) |
| 37 | \( 1 + (-5.87 + 1.59i)T \) |
good | 7 | \( 1 + (1.93 + 1.93i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.54iT - 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 - 3.60iT - 17T^{2} \) |
| 19 | \( 1 + (2.47 - 2.47i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.49T + 23T^{2} \) |
| 29 | \( 1 + (-4.58 - 4.58i)T + 29iT^{2} \) |
| 31 | \( 1 + (-7.33 + 7.33i)T - 31iT^{2} \) |
| 41 | \( 1 - 4.31iT - 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + (-4.51 - 4.51i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.301 - 0.301i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.813 - 0.813i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.867 - 0.867i)T - 61iT^{2} \) |
| 67 | \( 1 + (-5.77 + 5.77i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 + (3.70 + 3.70i)T + 73iT^{2} \) |
| 79 | \( 1 + (4.70 - 4.70i)T - 79iT^{2} \) |
| 83 | \( 1 + (0.524 - 0.524i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.69 - 6.69i)T + 89iT^{2} \) |
| 97 | \( 1 - 17.0iT - 97T^{2} \) |
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
−9.992226428734148444620823584522, −9.255067175434566753976156297279, −8.105066570380167218124144292199, −7.73520459104895045855928573608, −6.66452406989692662515313681850, −6.01424771435013194105998691234, −4.73933193201284209080512778474, −3.91561686405404642229820190652, −2.28630187303525833410070002051, −1.04496924522144556308978499585,
0.27387868310003873803496342688, 2.51143749145169355432869945298, 3.11026502119964757601590423709, 4.46345891479709751284429210814, 5.74590789340708953030920407493, 6.43228125210117704427710305070, 7.11818330580808154535077994613, 8.199716678053942195398937419630, 8.969635768390180860921478735568, 9.968385763427571268907453021151