L(s) = 1 | + (0.961 − 0.275i)2-s + (0.848 − 0.529i)4-s + (0.669 − 0.743i)8-s + (−0.241 + 0.970i)11-s + (0.241 + 0.970i)13-s + (0.438 − 0.898i)16-s + (−0.669 + 0.743i)17-s + (0.978 + 0.207i)19-s + (0.0348 + 0.999i)22-s + (0.559 − 0.829i)23-s + (0.5 + 0.866i)26-s + (−0.374 − 0.927i)29-s + (0.374 − 0.927i)31-s + (0.173 − 0.984i)32-s + (−0.438 + 0.898i)34-s + ⋯ |
L(s) = 1 | + (0.961 − 0.275i)2-s + (0.848 − 0.529i)4-s + (0.669 − 0.743i)8-s + (−0.241 + 0.970i)11-s + (0.241 + 0.970i)13-s + (0.438 − 0.898i)16-s + (−0.669 + 0.743i)17-s + (0.978 + 0.207i)19-s + (0.0348 + 0.999i)22-s + (0.559 − 0.829i)23-s + (0.5 + 0.866i)26-s + (−0.374 − 0.927i)29-s + (0.374 − 0.927i)31-s + (0.173 − 0.984i)32-s + (−0.438 + 0.898i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(5.059938807 - 0.04396295704i\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.059938807 - 0.04396295704i\) |
\(L(1)\) |
\(\approx\) |
\(2.010298210 - 0.2147873058i\) |
\(L(1)\) |
\(\approx\) |
\(2.010298210 - 0.2147873058i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.961 - 0.275i)T \) |
| 11 | \( 1 + (-0.241 + 0.970i)T \) |
| 13 | \( 1 + (0.241 + 0.970i)T \) |
| 17 | \( 1 + (-0.669 + 0.743i)T \) |
| 19 | \( 1 + (0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.559 - 0.829i)T \) |
| 29 | \( 1 + (-0.374 - 0.927i)T \) |
| 31 | \( 1 + (0.374 - 0.927i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.241 + 0.970i)T \) |
| 43 | \( 1 + (-0.939 - 0.342i)T \) |
| 47 | \( 1 + (0.374 + 0.927i)T \) |
| 53 | \( 1 + (0.669 + 0.743i)T \) |
| 59 | \( 1 + (0.719 - 0.694i)T \) |
| 61 | \( 1 + (0.241 - 0.970i)T \) |
| 67 | \( 1 + (-0.615 - 0.788i)T \) |
| 71 | \( 1 + (0.669 + 0.743i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.615 + 0.788i)T \) |
| 83 | \( 1 + (-0.848 - 0.529i)T \) |
| 89 | \( 1 + (0.104 + 0.994i)T \) |
| 97 | \( 1 + (0.615 - 0.788i)T \) |
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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
−17.90359087482442812070254688441, −17.25279349647503458043081505784, −16.36207500187627232220450933924, −15.90586548511494040430718733031, −15.36508459349671847196560357663, −14.64125690590058128619657829012, −13.69933162663554275917538294045, −13.53180394931867349638054294488, −12.800747791737802326640009106927, −11.8999486125537938541055787326, −11.416143235045154289626267774963, −10.71605501369058585339965069611, −10.03687174508037203436755497997, −8.80555949333161268465225077600, −8.440109019348652232812025640914, −7.32514583248659886484335514786, −7.05159172481539197781882165104, −6.034076709571829893854182726096, −5.24584402354749126552782999245, −5.07659732389798290659022531618, −3.78900414000884093827534101999, −3.222670868505889643905284580434, −2.687685270050433688045349095723, −1.53263374273575709502749927102, −0.60386937129498683573452404475,
0.725605711667608702955714410461, 1.800276318042384504786292658203, 2.24177193119745646402646975587, 3.2052162396068108580356000417, 4.12230433869682487942869208380, 4.51543251354551218729509166457, 5.348954937572721824350920048539, 6.18018597956040943725756499565, 6.780741001761375730668587038064, 7.45412037555988900805564331267, 8.3036303085656910868027506559, 9.37626865348054649720800353321, 9.878603646949736322939255353986, 10.72768212996754079237009496882, 11.35100799164115342994605636158, 11.97987765957086939715600303603, 12.67850551386814482492774983053, 13.238230061429131786190633853725, 13.95517788114173495339917065296, 14.54920708522438142005650420417, 15.32398187167668649354059237001, 15.67410250436120862039899820935, 16.618349830458184905079894582407, 17.13450519853733063359955354571, 18.114602269914891119640418396716