L(s) = 1 | + (0.998 − 0.0570i)2-s + (−0.309 − 0.951i)3-s + (0.993 − 0.113i)4-s + (0.466 − 0.884i)5-s + (−0.362 − 0.931i)6-s + (0.985 − 0.170i)8-s + (−0.809 + 0.587i)9-s + (0.415 − 0.909i)10-s + (−0.415 − 0.909i)12-s + (−0.870 − 0.491i)13-s + (−0.985 − 0.170i)15-s + (0.974 − 0.226i)16-s + (0.941 + 0.336i)17-s + (−0.774 + 0.633i)18-s + (−0.0285 − 0.999i)19-s + (0.362 − 0.931i)20-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0570i)2-s + (−0.309 − 0.951i)3-s + (0.993 − 0.113i)4-s + (0.466 − 0.884i)5-s + (−0.362 − 0.931i)6-s + (0.985 − 0.170i)8-s + (−0.809 + 0.587i)9-s + (0.415 − 0.909i)10-s + (−0.415 − 0.909i)12-s + (−0.870 − 0.491i)13-s + (−0.985 − 0.170i)15-s + (0.974 − 0.226i)16-s + (0.941 + 0.336i)17-s + (−0.774 + 0.633i)18-s + (−0.0285 − 0.999i)19-s + (0.362 − 0.931i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.246063463 - 2.254088339i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.246063463 - 2.254088339i\) |
\(L(1)\) |
\(\approx\) |
\(1.524177143 - 1.009022392i\) |
\(L(1)\) |
\(\approx\) |
\(1.524177143 - 1.009022392i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.998 - 0.0570i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.466 - 0.884i)T \) |
| 13 | \( 1 + (-0.870 - 0.491i)T \) |
| 17 | \( 1 + (0.941 + 0.336i)T \) |
| 19 | \( 1 + (-0.0285 - 0.999i)T \) |
| 23 | \( 1 + (-0.654 + 0.755i)T \) |
| 29 | \( 1 + (0.564 - 0.825i)T \) |
| 31 | \( 1 + (-0.198 - 0.980i)T \) |
| 37 | \( 1 + (-0.736 - 0.676i)T \) |
| 41 | \( 1 + (-0.254 + 0.967i)T \) |
| 43 | \( 1 + (0.142 - 0.989i)T \) |
| 47 | \( 1 + (-0.774 - 0.633i)T \) |
| 53 | \( 1 + (0.974 + 0.226i)T \) |
| 59 | \( 1 + (0.254 + 0.967i)T \) |
| 61 | \( 1 + (-0.998 - 0.0570i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (0.610 + 0.791i)T \) |
| 73 | \( 1 + (0.516 + 0.856i)T \) |
| 79 | \( 1 + (-0.696 - 0.717i)T \) |
| 83 | \( 1 + (0.0855 + 0.996i)T \) |
| 89 | \( 1 + (0.959 - 0.281i)T \) |
| 97 | \( 1 + (0.466 + 0.884i)T \) |
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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
−22.47865326626376606042923583705, −21.54998334586370923385061984852, −21.21848394438813586725559053532, −20.30606181662241654931011897811, −19.39166252560611821126917669417, −18.365059697873823054319060560031, −17.31302419254990503149040706586, −16.54271426533465180940762039388, −15.92395089657894895189464363413, −14.86742281309294666253894342332, −14.37035696650729615682875941211, −13.91994217322275908640346556637, −12.394427730080672712261351104751, −11.95270148915697976333134222312, −10.86763871147281095183079355861, −10.27847532030475306417845236432, −9.59617707978283118605215310871, −8.164729349331996612339940244775, −7.016530166402570871270417484219, −6.28615514580486276729546450374, −5.40638628731254030017411882315, −4.681636223511091195637723214204, −3.56464397204827531101213281541, −2.95278759479296947333515972573, −1.80134378490137452557809333362,
0.8322484659667577776520425396, 1.92909582848377649052915853656, 2.703499559071797957712146133370, 4.05854879279410680154261139032, 5.242407013490817267183083922635, 5.58113596696648703853078363618, 6.59252284720294553000192950976, 7.5436201006269224874528982950, 8.24850737136764180688024051432, 9.61627468209648533375514975713, 10.52757840250111465814244265064, 11.7676837064397748110886476278, 12.11324680921941948211459001045, 13.03818278275244493472114623498, 13.47518027356739657964788480369, 14.318291869250519375397499178411, 15.27337499311999528581138126331, 16.27208380959265407439240474532, 17.105619873776955891623579933356, 17.5641476955676107507967128368, 18.80541044885499221393482904583, 19.79701674380363571383178531169, 20.075980074547215583865151992864, 21.28697469319257866692788788163, 21.7935434402505360498731050661