L(s) = 1 | + (−0.909 + 0.415i)3-s + (0.540 − 0.841i)7-s + (0.654 − 0.755i)9-s + (0.142 + 0.989i)11-s + (−0.540 − 0.841i)13-s + (−0.281 − 0.959i)17-s + (−0.959 − 0.281i)19-s + (−0.142 + 0.989i)21-s + (−0.281 + 0.959i)27-s + (0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (−0.540 − 0.841i)33-s + (−0.755 − 0.654i)37-s + (0.841 + 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯ |
L(s) = 1 | + (−0.909 + 0.415i)3-s + (0.540 − 0.841i)7-s + (0.654 − 0.755i)9-s + (0.142 + 0.989i)11-s + (−0.540 − 0.841i)13-s + (−0.281 − 0.959i)17-s + (−0.959 − 0.281i)19-s + (−0.142 + 0.989i)21-s + (−0.281 + 0.959i)27-s + (0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (−0.540 − 0.841i)33-s + (−0.755 − 0.654i)37-s + (0.841 + 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5615215109 - 0.4894430738i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5615215109 - 0.4894430738i\) |
\(L(1)\) |
\(\approx\) |
\(0.7553847396 - 0.1064286228i\) |
\(L(1)\) |
\(\approx\) |
\(0.7553847396 - 0.1064286228i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.909 + 0.415i)T \) |
| 7 | \( 1 + (0.540 - 0.841i)T \) |
| 11 | \( 1 + (0.142 + 0.989i)T \) |
| 13 | \( 1 + (-0.540 - 0.841i)T \) |
| 17 | \( 1 + (-0.281 - 0.959i)T \) |
| 19 | \( 1 + (-0.959 - 0.281i)T \) |
| 29 | \( 1 + (0.959 - 0.281i)T \) |
| 31 | \( 1 + (-0.415 + 0.909i)T \) |
| 37 | \( 1 + (-0.755 - 0.654i)T \) |
| 41 | \( 1 + (-0.654 - 0.755i)T \) |
| 43 | \( 1 + (-0.909 + 0.415i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.540 - 0.841i)T \) |
| 59 | \( 1 + (0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (0.989 + 0.142i)T \) |
| 71 | \( 1 + (0.142 - 0.989i)T \) |
| 73 | \( 1 + (-0.281 + 0.959i)T \) |
| 79 | \( 1 + (0.841 - 0.540i)T \) |
| 83 | \( 1 + (-0.755 - 0.654i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (0.755 - 0.654i)T \) |
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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
−24.022288204090236699551285874741, −23.52908443387227820699112995145, −22.18927019187190977730280611497, −21.75596107336139225272372748271, −21.05241395433599494458442342344, −19.479253731661876543244293315557, −18.92853865852467291581999991225, −18.15534209153328652825977860170, −17.13751220241056540938489858109, −16.62759542081012740724656824223, −15.51577677117336634474032344381, −14.5983843112129793795539023296, −13.55053065483663714514874997317, −12.55658905571232840245305290849, −11.79030540590359602781655617928, −11.1092056810961321816508321295, −10.15369765230076447384887200154, −8.79074332032722880849551144267, −8.0822625090000312489427203287, −6.73788624353764195381897589616, −6.04458182028619933329607107879, −5.10525625633341107732219319640, −4.08493542723954739046163647652, −2.41438148198022496746674602762, −1.416667814541295911078628413852,
0.48358511801188278757130992381, 1.98020249721350247426133142005, 3.60978193169991281221134812350, 4.71571581749245719728456813337, 5.18402554127843103936776941832, 6.7135324669680756573867652246, 7.242999210383454434621165172879, 8.56339562434148693133592581049, 9.89688025462214002948143441296, 10.39167266723326521619402536134, 11.34427245511331635047156292413, 12.22840667984128419707134660998, 13.07433909211891999802924856473, 14.29549159906689075392297985470, 15.16211528808827266969045636547, 15.976209660657142618343978449031, 17.062861872336738810412396438459, 17.55692068431521945285496029671, 18.216521226285472685727317851098, 19.66151502718228703245225715790, 20.39414272881567050565065854566, 21.20495031502734572695244909883, 22.1305390127486202431748146596, 23.0296841770718735215739175001, 23.40275003217271424562075950993