L(s) = 1 | + (0.445 − 0.895i)2-s + (−0.739 + 0.673i)3-s + (−0.602 − 0.798i)4-s + (−0.445 − 0.895i)5-s + (0.273 + 0.961i)6-s + (0.932 + 0.361i)7-s + (−0.982 + 0.183i)8-s + (0.0922 − 0.995i)9-s − 10-s + (−0.602 − 0.798i)11-s + (0.982 + 0.183i)12-s + (−0.932 − 0.361i)13-s + (0.739 − 0.673i)14-s + (0.932 + 0.361i)15-s + (−0.273 + 0.961i)16-s + (−0.982 − 0.183i)17-s + ⋯ |
L(s) = 1 | + (0.445 − 0.895i)2-s + (−0.739 + 0.673i)3-s + (−0.602 − 0.798i)4-s + (−0.445 − 0.895i)5-s + (0.273 + 0.961i)6-s + (0.932 + 0.361i)7-s + (−0.982 + 0.183i)8-s + (0.0922 − 0.995i)9-s − 10-s + (−0.602 − 0.798i)11-s + (0.982 + 0.183i)12-s + (−0.932 − 0.361i)13-s + (0.739 − 0.673i)14-s + (0.932 + 0.361i)15-s + (−0.273 + 0.961i)16-s + (−0.982 − 0.183i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1513586974 - 0.6753640846i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1513586974 - 0.6753640846i\) |
\(L(1)\) |
\(\approx\) |
\(0.6369017687 - 0.5001221540i\) |
\(L(1)\) |
\(\approx\) |
\(0.6369017687 - 0.5001221540i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (0.445 - 0.895i)T \) |
| 3 | \( 1 + (-0.739 + 0.673i)T \) |
| 5 | \( 1 + (-0.445 - 0.895i)T \) |
| 7 | \( 1 + (0.932 + 0.361i)T \) |
| 11 | \( 1 + (-0.602 - 0.798i)T \) |
| 13 | \( 1 + (-0.932 - 0.361i)T \) |
| 17 | \( 1 + (-0.982 - 0.183i)T \) |
| 19 | \( 1 + (-0.850 - 0.526i)T \) |
| 23 | \( 1 + (0.273 - 0.961i)T \) |
| 29 | \( 1 + (0.273 - 0.961i)T \) |
| 31 | \( 1 + (0.850 - 0.526i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (0.850 + 0.526i)T \) |
| 47 | \( 1 + (-0.0922 + 0.995i)T \) |
| 53 | \( 1 + (0.850 + 0.526i)T \) |
| 59 | \( 1 + (0.0922 - 0.995i)T \) |
| 61 | \( 1 + (0.0922 + 0.995i)T \) |
| 67 | \( 1 + (-0.932 - 0.361i)T \) |
| 71 | \( 1 + (0.602 - 0.798i)T \) |
| 73 | \( 1 + (0.932 - 0.361i)T \) |
| 79 | \( 1 + (-0.739 - 0.673i)T \) |
| 83 | \( 1 + (0.982 - 0.183i)T \) |
| 89 | \( 1 + (-0.445 - 0.895i)T \) |
| 97 | \( 1 + (0.602 + 0.798i)T \) |
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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
−29.10993191522478339097341276728, −27.598551113494290460722870996099, −26.932016442513727856874850756941, −25.80311352667810361476304369539, −24.73931687104254504729395338747, −23.67120953198372941706347054732, −23.336570389707936840908592459585, −22.242987772354623163063183854600, −21.38273913510089021747900580543, −19.67842803589765341739169089996, −18.38092768770199368299514126825, −17.698725089192260833017816073099, −16.900785406380537235922655441329, −15.49682255022680099472907512220, −14.663014373621312690111976465016, −13.61624631782712055349332876556, −12.431267603777397155479652815686, −11.48153728971466701972950209840, −10.31444008345737643818442244142, −8.268228280339021584886217378358, −7.309839933676483029519705955745, −6.7246728660476574503606664209, −5.22623961443522521695501853983, −4.26351387102396963698181738345, −2.28734230030178440858642385822,
0.57677832911037114199079512212, 2.56041308522518871979001300001, 4.41170226522358660109056270833, 4.829226543072681806589010944660, 6.01193194099715327608082793358, 8.26758709888365899789911528398, 9.27723013960039803657988648169, 10.62694251174617650184847559682, 11.41864023323878065296540314817, 12.26759318562946933230449149618, 13.31241823536522809969769698582, 14.87586194423071207589839409853, 15.60355333778649379099858271737, 16.99946781459529550703228480302, 17.91084830766463633480821793842, 19.19010502503936210277485562777, 20.38431404348535445610002777071, 21.12001327849986418100023226855, 21.863325805848936283282943053210, 22.92506559017504025374303406382, 24.031862042009813243993124393261, 24.45812266650645506746897398965, 26.81163943561112619248431383570, 27.23049146513384562276326063393, 28.315928433349219724520514235157