L(s) = 1 | + (0.0257 + 0.999i)2-s + (0.328 + 0.944i)3-s + (−0.998 + 0.0514i)4-s + (0.423 + 0.905i)5-s + (−0.935 + 0.352i)6-s + (−0.376 + 0.926i)7-s + (−0.0771 − 0.997i)8-s + (−0.784 + 0.620i)9-s + (−0.894 + 0.447i)10-s + (0.600 + 0.799i)11-s + (−0.376 − 0.926i)12-s + (−0.784 + 0.620i)13-s + (−0.935 − 0.352i)14-s + (−0.716 + 0.697i)15-s + (0.994 − 0.102i)16-s + (0.870 − 0.492i)17-s + ⋯ |
L(s) = 1 | + (0.0257 + 0.999i)2-s + (0.328 + 0.944i)3-s + (−0.998 + 0.0514i)4-s + (0.423 + 0.905i)5-s + (−0.935 + 0.352i)6-s + (−0.376 + 0.926i)7-s + (−0.0771 − 0.997i)8-s + (−0.784 + 0.620i)9-s + (−0.894 + 0.447i)10-s + (0.600 + 0.799i)11-s + (−0.376 − 0.926i)12-s + (−0.784 + 0.620i)13-s + (−0.935 − 0.352i)14-s + (−0.716 + 0.697i)15-s + (0.994 − 0.102i)16-s + (0.870 − 0.492i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4113384498 + 1.155171880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4113384498 + 1.155171880i\) |
\(L(1)\) |
\(\approx\) |
\(0.4332995235 + 0.9843133438i\) |
\(L(1)\) |
\(\approx\) |
\(0.4332995235 + 0.9843133438i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 367 | \( 1 \) |
good | 2 | \( 1 + (0.0257 + 0.999i)T \) |
| 3 | \( 1 + (0.328 + 0.944i)T \) |
| 5 | \( 1 + (0.423 + 0.905i)T \) |
| 7 | \( 1 + (-0.376 + 0.926i)T \) |
| 11 | \( 1 + (0.600 + 0.799i)T \) |
| 13 | \( 1 + (-0.784 + 0.620i)T \) |
| 17 | \( 1 + (0.870 - 0.492i)T \) |
| 19 | \( 1 + (-0.179 - 0.983i)T \) |
| 23 | \( 1 + (0.994 + 0.102i)T \) |
| 29 | \( 1 + (0.978 + 0.204i)T \) |
| 31 | \( 1 + (-0.998 + 0.0514i)T \) |
| 37 | \( 1 + (0.128 - 0.991i)T \) |
| 41 | \( 1 + (0.600 - 0.799i)T \) |
| 43 | \( 1 + (-0.179 + 0.983i)T \) |
| 47 | \( 1 + (-0.558 - 0.829i)T \) |
| 53 | \( 1 + (0.423 + 0.905i)T \) |
| 59 | \( 1 + (0.952 - 0.304i)T \) |
| 61 | \( 1 + (-0.935 - 0.352i)T \) |
| 67 | \( 1 + (0.952 + 0.304i)T \) |
| 71 | \( 1 + (-0.998 - 0.0514i)T \) |
| 73 | \( 1 + (0.916 + 0.400i)T \) |
| 79 | \( 1 + (-0.716 - 0.697i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.558 - 0.829i)T \) |
| 97 | \( 1 + (0.994 + 0.102i)T \) |
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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
−23.99954700076214623130500237580, −23.30089006982383293236170212507, −22.383504821780849170987734642193, −21.20684191873321614339584231533, −20.519586094615659650053423998825, −19.6321944847981591356846560320, −19.24693373512644410058477691409, −18.09787375887001770764237832487, −17.04801336895885372519301395460, −16.73989613011425426736241249622, −14.64124531022514212669123745378, −13.97375719183538157995502030386, −13.06937186954720914432406347763, −12.58117349547644600079155932387, −11.67962528535194921320703855502, −10.39868440200196433401026118001, −9.54966432419187604505263534354, −8.52554225879915875110409300512, −7.76387183961811203947639851047, −6.285276108317610729117999191906, −5.240149241262773099565836872108, −3.86721986552639366106137865672, −2.930028862782833868765992826074, −1.51094882519308920767810926926, −0.76717049189605581314895837678,
2.38936002581567615241235137188, 3.42032679923624969845681799543, 4.69949234508469130665268410337, 5.51947817501986802804664699582, 6.663325678080210494574628378905, 7.46243084701435262870268875894, 9.01519186064774541792019957567, 9.382125858112019603912852747893, 10.25315064043024538571791568540, 11.601125313066231999783237285716, 12.809779387822536268132364190033, 14.11291503084768006305545020668, 14.657398112043856956186826540966, 15.258475659030297096667432124170, 16.17161019589370002922407384645, 17.104721398825555356863333130654, 17.932067684907171483002387382179, 18.978416730105450530770960431075, 19.70754593082965649806645760666, 21.40840637490867241349164998401, 21.71662093497018506318090928719, 22.60762393904665562265626470527, 23.188716443990366196642517937460, 24.74694836952202003661218672132, 25.32505907733406904585382715669