L(s) = 1 | + (−0.438 + 0.898i)2-s + (−0.848 − 0.529i)3-s + (−0.615 − 0.788i)4-s + (0.848 − 0.529i)6-s + (−0.5 + 0.866i)7-s + (0.978 − 0.207i)8-s + (0.438 + 0.898i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (0.997 + 0.0697i)13-s + (−0.559 − 0.829i)14-s + (−0.241 + 0.970i)16-s + (0.990 + 0.139i)17-s − 18-s + (0.882 − 0.469i)21-s + (−0.848 − 0.529i)22-s + ⋯ |
L(s) = 1 | + (−0.438 + 0.898i)2-s + (−0.848 − 0.529i)3-s + (−0.615 − 0.788i)4-s + (0.848 − 0.529i)6-s + (−0.5 + 0.866i)7-s + (0.978 − 0.207i)8-s + (0.438 + 0.898i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (0.997 + 0.0697i)13-s + (−0.559 − 0.829i)14-s + (−0.241 + 0.970i)16-s + (0.990 + 0.139i)17-s − 18-s + (0.882 − 0.469i)21-s + (−0.848 − 0.529i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4594636467 + 0.8364077354i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4594636467 + 0.8364077354i\) |
\(L(1)\) |
\(\approx\) |
\(0.5939592693 + 0.2970224419i\) |
\(L(1)\) |
\(\approx\) |
\(0.5939592693 + 0.2970224419i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.438 + 0.898i)T \) |
| 3 | \( 1 + (-0.848 - 0.529i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.104 + 0.994i)T \) |
| 13 | \( 1 + (0.997 + 0.0697i)T \) |
| 17 | \( 1 + (0.990 + 0.139i)T \) |
| 23 | \( 1 + (0.961 + 0.275i)T \) |
| 29 | \( 1 + (-0.990 + 0.139i)T \) |
| 31 | \( 1 + (-0.669 - 0.743i)T \) |
| 37 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.241 - 0.970i)T \) |
| 43 | \( 1 + (-0.939 - 0.342i)T \) |
| 47 | \( 1 + (0.990 - 0.139i)T \) |
| 53 | \( 1 + (0.615 + 0.788i)T \) |
| 59 | \( 1 + (0.719 + 0.694i)T \) |
| 61 | \( 1 + (0.961 + 0.275i)T \) |
| 67 | \( 1 + (0.882 + 0.469i)T \) |
| 71 | \( 1 + (-0.0348 + 0.999i)T \) |
| 73 | \( 1 + (-0.997 + 0.0697i)T \) |
| 79 | \( 1 + (-0.848 - 0.529i)T \) |
| 83 | \( 1 + (0.669 + 0.743i)T \) |
| 89 | \( 1 + (0.241 + 0.970i)T \) |
| 97 | \( 1 + (0.882 - 0.469i)T \) |
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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
−23.1845425303580060160067378715, −22.37513538936314970864712285630, −21.488314557729545282414135474860, −20.843361901478542926645180077554, −20.069693848008437144005822958529, −18.89230897339597679184741097151, −18.377918021269139576354782840223, −17.24294418223344592245767318987, −16.54561459294239752226361625087, −16.09656176018870375961744619596, −14.56278017307449243302019758762, −13.37345614449666547886791102914, −12.8026424314019141048851301598, −11.542338340461018881475362138948, −11.00756582928090690161676355607, −10.248966409701613379606581227639, −9.43631406114998074052647463343, −8.405692086242841863875399071578, −7.201744496400168467328490450696, −6.04750604422963527230417903534, −4.925040656411413500951883460841, −3.716738376698378263728484612988, −3.23137402456926993604535256978, −1.222386385950155911253481889870, −0.47967686260256330687009778984,
0.92891916504029128984970083906, 2.10604819138487740035755937256, 3.99317354625360070032441589997, 5.41568979463715967136246604862, 5.759069543910615720408905307988, 6.89107058300657515014464439317, 7.55457652893914598996701604601, 8.72273026483245284800907974428, 9.63940287694895573692276291073, 10.593444927220215164667632618129, 11.64552691492403969207128784791, 12.74634883126726827298829967780, 13.30939989881658155320293331936, 14.65551269365279900001649448519, 15.45421476156864089242832440706, 16.289270668592391035380593207987, 17.00736719236122905556211871006, 17.91098696059769969196946264868, 18.66156456806331348178588714485, 19.047437249486517243986766254932, 20.36999090788672308764769346397, 21.67165730674354489387130031610, 22.579025055057808881054066162791, 23.20077447929719423184390794562, 23.80329010886051605768865921568