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 | \( 1 + 17.0iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.968385763427571268907453021151, −8.969635768390180860921478735568, −8.199716678053942195398937419630, −7.11818330580808154535077994613, −6.43228125210117704427710305070, −5.74590789340708953030920407493, −4.46345891479709751284429210814, −3.11026502119964757601590423709, −2.51143749145169355432869945298, −0.27387868310003873803496342688,
1.04496924522144556308978499585, 2.28630187303525833410070002051, 3.91561686405404642229820190652, 4.73933193201284209080512778474, 6.01424771435013194105998691234, 6.66452406989692662515313681850, 7.73520459104895045855928573608, 8.105066570380167218124144292199, 9.255067175434566753976156297279, 9.992226428734148444620823584522