L(s) = 1 | + (−0.654 − 0.755i)3-s + (−0.909 − 0.415i)7-s + (−0.142 + 0.989i)9-s + (0.281 − 0.959i)11-s + (−0.415 − 0.909i)13-s + (0.540 − 0.841i)17-s + (0.540 + 0.841i)19-s + (0.281 + 0.959i)21-s + (0.841 − 0.540i)27-s + (0.540 − 0.841i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.142 − 0.989i)37-s + (−0.415 + 0.909i)39-s + (0.142 + 0.989i)41-s + ⋯ |
L(s) = 1 | + (−0.654 − 0.755i)3-s + (−0.909 − 0.415i)7-s + (−0.142 + 0.989i)9-s + (0.281 − 0.959i)11-s + (−0.415 − 0.909i)13-s + (0.540 − 0.841i)17-s + (0.540 + 0.841i)19-s + (0.281 + 0.959i)21-s + (0.841 − 0.540i)27-s + (0.540 − 0.841i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (0.142 − 0.989i)37-s + (−0.415 + 0.909i)39-s + (0.142 + 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2340847796 - 0.8864612646i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2340847796 - 0.8864612646i\) |
\(L(1)\) |
\(\approx\) |
\(0.6943617827 - 0.3740687859i\) |
\(L(1)\) |
\(\approx\) |
\(0.6943617827 - 0.3740687859i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.909 - 0.415i)T \) |
| 11 | \( 1 + (0.281 - 0.959i)T \) |
| 13 | \( 1 + (-0.415 - 0.909i)T \) |
| 17 | \( 1 + (0.540 - 0.841i)T \) |
| 19 | \( 1 + (0.540 + 0.841i)T \) |
| 29 | \( 1 + (0.540 - 0.841i)T \) |
| 31 | \( 1 + (0.654 - 0.755i)T \) |
| 37 | \( 1 + (0.142 - 0.989i)T \) |
| 41 | \( 1 + (0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.654 + 0.755i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (0.909 - 0.415i)T \) |
| 61 | \( 1 + (-0.755 - 0.654i)T \) |
| 67 | \( 1 + (0.959 - 0.281i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (-0.540 - 0.841i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.142 + 0.989i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (-0.989 + 0.142i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.473238885377434113637834446903, −19.729705834996993384981294591307, −19.04933998375034871445533120067, −18.129880318805184279363158232395, −17.364014375551985435233679782258, −16.78804436069615072355837232815, −16.02509468037188754868854831570, −15.413601766158990632422750870475, −14.75938342149434538095965034961, −13.88258715087388680360211302845, −12.79763435429706425716941748561, −12.082212232554726473560619436040, −11.7350894854924779715110423786, −10.45054158541196654863764454113, −10.08174962913239060989833673887, −9.20308397370991645561182139324, −8.73221247741433337276169065946, −7.18071998010812054485734656470, −6.71868307906411533333081177045, −5.82205870102010025206922308904, −4.997250164100016750566553821230, −4.23907548953694933903202160999, −3.396536040728746803534577415931, −2.43324035395806841025850723457, −1.14467124484306481827959132385,
0.44859084701094896658143768340, 1.11861431445095151029160536943, 2.57486547555078115865152114021, 3.22781473233150632515156167079, 4.34697091863107856514409498118, 5.501302469435631263643265393056, 5.991507122584851622885051858822, 6.7646122328096143022642351719, 7.69731591012010266940368485714, 8.1301086734664915555419031375, 9.50173916285419383962044992912, 10.05113151871378746859276370898, 10.98021068609801078369519263022, 11.68287648051528205297395810391, 12.44688901655714769375164189956, 13.07800990876327112584986741701, 13.81037443951763427841133931736, 14.412009200575650639388801125357, 15.71710388732716916777231352995, 16.31420319410416243065264196080, 16.853179453774449972331237720233, 17.66835006123175779122604009325, 18.37120201550187018782934364186, 19.19484329930901549445138431133, 19.53148116254147919245190721469