L(s) = 1 | + (−0.775 − 0.631i)2-s + (−0.775 − 0.631i)3-s + (0.203 + 0.979i)4-s + (0.854 − 0.519i)5-s + (0.203 + 0.979i)6-s + (0.203 − 0.979i)7-s + (0.460 − 0.887i)8-s + (0.203 + 0.979i)9-s + (−0.990 − 0.136i)10-s + (0.962 − 0.269i)11-s + (0.460 − 0.887i)12-s + (−0.990 + 0.136i)13-s + (−0.775 + 0.631i)14-s + (−0.990 − 0.136i)15-s + (−0.917 + 0.398i)16-s + (−0.990 + 0.136i)17-s + ⋯ |
L(s) = 1 | + (−0.775 − 0.631i)2-s + (−0.775 − 0.631i)3-s + (0.203 + 0.979i)4-s + (0.854 − 0.519i)5-s + (0.203 + 0.979i)6-s + (0.203 − 0.979i)7-s + (0.460 − 0.887i)8-s + (0.203 + 0.979i)9-s + (−0.990 − 0.136i)10-s + (0.962 − 0.269i)11-s + (0.460 − 0.887i)12-s + (−0.990 + 0.136i)13-s + (−0.775 + 0.631i)14-s + (−0.990 − 0.136i)15-s + (−0.917 + 0.398i)16-s + (−0.990 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.005727456793 - 0.6429317513i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.005727456793 - 0.6429317513i\) |
\(L(1)\) |
\(\approx\) |
\(0.4690956667 - 0.4333814784i\) |
\(L(1)\) |
\(\approx\) |
\(0.4690956667 - 0.4333814784i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.775 - 0.631i)T \) |
| 3 | \( 1 + (-0.775 - 0.631i)T \) |
| 5 | \( 1 + (0.854 - 0.519i)T \) |
| 7 | \( 1 + (0.203 - 0.979i)T \) |
| 11 | \( 1 + (0.962 - 0.269i)T \) |
| 13 | \( 1 + (-0.990 + 0.136i)T \) |
| 17 | \( 1 + (-0.990 + 0.136i)T \) |
| 19 | \( 1 + (-0.775 - 0.631i)T \) |
| 29 | \( 1 + (0.962 - 0.269i)T \) |
| 31 | \( 1 + (0.682 + 0.730i)T \) |
| 37 | \( 1 + (-0.917 - 0.398i)T \) |
| 41 | \( 1 + (-0.0682 - 0.997i)T \) |
| 43 | \( 1 + (-0.576 - 0.816i)T \) |
| 47 | \( 1 + (0.682 - 0.730i)T \) |
| 53 | \( 1 + (-0.990 + 0.136i)T \) |
| 59 | \( 1 + (-0.775 + 0.631i)T \) |
| 61 | \( 1 + (0.854 - 0.519i)T \) |
| 67 | \( 1 + (0.962 - 0.269i)T \) |
| 71 | \( 1 + (-0.0682 + 0.997i)T \) |
| 73 | \( 1 + (0.962 + 0.269i)T \) |
| 79 | \( 1 + (-0.576 - 0.816i)T \) |
| 83 | \( 1 + (0.854 - 0.519i)T \) |
| 89 | \( 1 + (-0.334 - 0.942i)T \) |
| 97 | \( 1 + (-0.0682 + 0.997i)T \) |
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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
−24.141002300036613100457342751087, −22.90679474854566059338621989775, −22.2620210323087146311392146572, −21.61392673211256776548432059882, −20.60195980236466732343882061038, −19.44239032523908543469015102090, −18.57597926258437063500717910946, −17.62928261813872421565036241459, −17.37576473017661169072895797287, −16.45946596930055950181030865541, −15.336992874816840904459669001780, −14.89256630408580737025829273399, −14.069486893469197014212358112040, −12.50668887482924663927489257212, −11.55505023915553962121907548302, −10.71444459192700996605300182502, −9.74328770695239031923898798552, −9.34451753750536985668190767990, −8.24943238527486910305949778755, −6.65164252423378008705741385519, −6.39758337531976160624349350253, −5.33085224860207439176467011049, −4.52298700300367274954179713825, −2.64573493331438631218683900235, −1.5470850127084180222175567479,
0.4983028217385308085122671906, 1.55035227341893251881842221120, 2.390856990736522608003617637148, 4.13529640087704708723557545693, 4.9970077833155917468817818723, 6.60527056865706440125254904731, 6.92462165799894543725820396665, 8.2587605921114226759193804277, 9.105319324057063618151207530929, 10.218995057729422554149294202590, 10.78120304080219637299475461378, 11.833080201194038633550750572836, 12.52024439384514264644987355767, 13.45129534479594920568469953241, 14.07247281367401407517620001592, 15.85095171621752383863946586791, 16.89485793708082437193679105275, 17.431254288232365282734536401263, 17.53112101765794466896495117493, 18.93313415844127023595594344632, 19.66307247600072115600923934497, 20.292380831339472927214099284, 21.566051460179124774629815556814, 21.91550969705300965058723756242, 22.9600091488751908920610180439