L(s) = 1 | + (0.328 + 0.944i)2-s + (−0.935 − 0.352i)3-s + (−0.784 + 0.620i)4-s + (−0.558 + 0.829i)5-s + (0.0257 − 0.999i)6-s + (0.952 + 0.304i)7-s + (−0.843 − 0.536i)8-s + (0.751 + 0.660i)9-s + (−0.967 − 0.254i)10-s + (0.870 − 0.492i)11-s + (0.952 − 0.304i)12-s + (0.751 + 0.660i)13-s + (0.0257 + 0.999i)14-s + (0.815 − 0.579i)15-s + (0.229 − 0.973i)16-s + (0.916 − 0.400i)17-s + ⋯ |
L(s) = 1 | + (0.328 + 0.944i)2-s + (−0.935 − 0.352i)3-s + (−0.784 + 0.620i)4-s + (−0.558 + 0.829i)5-s + (0.0257 − 0.999i)6-s + (0.952 + 0.304i)7-s + (−0.843 − 0.536i)8-s + (0.751 + 0.660i)9-s + (−0.967 − 0.254i)10-s + (0.870 − 0.492i)11-s + (0.952 − 0.304i)12-s + (0.751 + 0.660i)13-s + (0.0257 + 0.999i)14-s + (0.815 − 0.579i)15-s + (0.229 − 0.973i)16-s + (0.916 − 0.400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3116725604 + 0.9552887120i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3116725604 + 0.9552887120i\) |
\(L(1)\) |
\(\approx\) |
\(0.6931129294 + 0.5948910480i\) |
\(L(1)\) |
\(\approx\) |
\(0.6931129294 + 0.5948910480i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 367 | \( 1 \) |
good | 2 | \( 1 + (0.328 + 0.944i)T \) |
| 3 | \( 1 + (-0.935 - 0.352i)T \) |
| 5 | \( 1 + (-0.558 + 0.829i)T \) |
| 7 | \( 1 + (0.952 + 0.304i)T \) |
| 11 | \( 1 + (0.870 - 0.492i)T \) |
| 13 | \( 1 + (0.751 + 0.660i)T \) |
| 17 | \( 1 + (0.916 - 0.400i)T \) |
| 19 | \( 1 + (-0.716 + 0.697i)T \) |
| 23 | \( 1 + (0.229 + 0.973i)T \) |
| 29 | \( 1 + (-0.894 + 0.447i)T \) |
| 31 | \( 1 + (-0.784 + 0.620i)T \) |
| 37 | \( 1 + (0.994 + 0.102i)T \) |
| 41 | \( 1 + (0.870 + 0.492i)T \) |
| 43 | \( 1 + (-0.716 - 0.697i)T \) |
| 47 | \( 1 + (-0.988 - 0.153i)T \) |
| 53 | \( 1 + (-0.558 + 0.829i)T \) |
| 59 | \( 1 + (-0.640 + 0.767i)T \) |
| 61 | \( 1 + (0.0257 + 0.999i)T \) |
| 67 | \( 1 + (-0.640 - 0.767i)T \) |
| 71 | \( 1 + (-0.784 - 0.620i)T \) |
| 73 | \( 1 + (0.600 - 0.799i)T \) |
| 79 | \( 1 + (0.815 + 0.579i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.988 - 0.153i)T \) |
| 97 | \( 1 + (0.229 + 0.973i)T \) |
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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
−23.91463414430098328510018433945, −23.323252990107910642796791868516, −22.614032335236892872381755651832, −21.58783614851309321242785942176, −20.7548901060498774396821487329, −20.296089801684382272163629511728, −19.15611394590943896439802079707, −18.09421213108134211550636024609, −17.28612013900346596034165128726, −16.54690579086297636121749588101, −15.18417870972523055467035535835, −14.58617677696937631034256732421, −13.01795768977638534811636789463, −12.533957546563733378410680031353, −11.39824480039197236532105604291, −11.10126108845177185550980585440, −9.91880936417172690325507183210, −8.923756869676740963799160140905, −7.83990832453955345609508012661, −6.20889350984765948790946327680, −5.156635293813744882656887536514, −4.38185954015165311830351767993, −3.701664471548380070765932723501, −1.678297242673048053064402173282, −0.72345814987930706715086790379,
1.47197165673798952919471849474, 3.4677131224934001536990006404, 4.41436534595780523515115737918, 5.61813122237462000058180493491, 6.330149908371150867512887424944, 7.315766335112479124676716881456, 8.03555173777666641348699198920, 9.244568260857198794633790724200, 10.80754760040844661358822539470, 11.58465352876240243749939717376, 12.243021632577518100848061499428, 13.575676273591764185059394620281, 14.43717533635237899767531699891, 15.12757188574554065636388187065, 16.34479030914467943295967375217, 16.80307469438296448018160673391, 18.03086194747281515807769891102, 18.453209476979338036347446921852, 19.31416873956380843467742844107, 21.16319638333296333883589368957, 21.76617773217671903939165115628, 22.63650433360870171443266114567, 23.50932679389345794143729022174, 23.81128659836510364605283849435, 24.9312523009537539745124390695