L(s) = 1 | + (−0.0855 − 0.996i)2-s + (−0.309 + 0.951i)3-s + (−0.985 + 0.170i)4-s + (0.998 − 0.0570i)5-s + (0.974 + 0.226i)6-s + (0.254 + 0.967i)8-s + (−0.809 − 0.587i)9-s + (−0.142 − 0.989i)10-s + (0.142 − 0.989i)12-s + (0.696 − 0.717i)13-s + (−0.254 + 0.967i)15-s + (0.941 − 0.336i)16-s + (−0.870 − 0.491i)17-s + (−0.516 + 0.856i)18-s + (−0.736 − 0.676i)19-s + (−0.974 + 0.226i)20-s + ⋯ |
L(s) = 1 | + (−0.0855 − 0.996i)2-s + (−0.309 + 0.951i)3-s + (−0.985 + 0.170i)4-s + (0.998 − 0.0570i)5-s + (0.974 + 0.226i)6-s + (0.254 + 0.967i)8-s + (−0.809 − 0.587i)9-s + (−0.142 − 0.989i)10-s + (0.142 − 0.989i)12-s + (0.696 − 0.717i)13-s + (−0.254 + 0.967i)15-s + (0.941 − 0.336i)16-s + (−0.870 − 0.491i)17-s + (−0.516 + 0.856i)18-s + (−0.736 − 0.676i)19-s + (−0.974 + 0.226i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5062815714 - 0.7991472919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5062815714 - 0.7991472919i\) |
\(L(1)\) |
\(\approx\) |
\(0.8098848188 - 0.3275630170i\) |
\(L(1)\) |
\(\approx\) |
\(0.8098848188 - 0.3275630170i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.0855 - 0.996i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.998 - 0.0570i)T \) |
| 13 | \( 1 + (0.696 - 0.717i)T \) |
| 17 | \( 1 + (-0.870 - 0.491i)T \) |
| 19 | \( 1 + (-0.736 - 0.676i)T \) |
| 23 | \( 1 + (-0.959 - 0.281i)T \) |
| 29 | \( 1 + (-0.993 - 0.113i)T \) |
| 31 | \( 1 + (0.466 - 0.884i)T \) |
| 37 | \( 1 + (0.897 - 0.441i)T \) |
| 41 | \( 1 + (-0.921 + 0.389i)T \) |
| 43 | \( 1 + (-0.841 + 0.540i)T \) |
| 47 | \( 1 + (-0.516 - 0.856i)T \) |
| 53 | \( 1 + (0.941 + 0.336i)T \) |
| 59 | \( 1 + (0.921 + 0.389i)T \) |
| 61 | \( 1 + (0.0855 - 0.996i)T \) |
| 67 | \( 1 + (-0.654 - 0.755i)T \) |
| 71 | \( 1 + (0.198 + 0.980i)T \) |
| 73 | \( 1 + (-0.0285 - 0.999i)T \) |
| 79 | \( 1 + (0.362 + 0.931i)T \) |
| 83 | \( 1 + (0.610 - 0.791i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (0.998 + 0.0570i)T \) |
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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
−22.42674897124833535092624538762, −21.91727528813342546206864539305, −20.90725211154534349376864335540, −19.68162042857030769501464083031, −18.812468147671992413780284029422, −18.1924349832090792017436140568, −17.59428160271721549377970747520, −16.836647421022139532117541751104, −16.239150700053921184064783722397, −14.98941579359460733635351134475, −14.221981804588894692464979386689, −13.45207570448902639430224528205, −13.0596044897920702710051332187, −11.97675333886667081147245216582, −10.81082595916093712983598240147, −9.93170319893057240804339205801, −8.81007000454869784638155211320, −8.30106936653753122282732974567, −7.1247330711697456273245024674, −6.37905384845733408820989959142, −5.95051148592437023601096359868, −4.96250133264390435842543136564, −3.76958423782505745383327425630, −2.13221855712711873295031942937, −1.340384824219674900417476225520,
0.467754992402125765960044013222, 1.978553036000110168704764254690, 2.84091417270407690102461164711, 3.91450429560428159065137673781, 4.75816601033682893251512019987, 5.60810648400600882394382283701, 6.44777716604685769194519100826, 8.196867939725118243455488017750, 8.97376756181227777351957540509, 9.69360451251058083202846611165, 10.37564845816469655961333093784, 11.08055516281541272788407055758, 11.80305863329354224142682152860, 13.0903801088561770766809635163, 13.44263973440562705975194594333, 14.55386282613649023904571627240, 15.31972499366930800979286437622, 16.48550386402430500799221216214, 17.17893939481645804168515492698, 17.972839489071485464990901705635, 18.442302217847668970688116776733, 19.91008597936269137846322483026, 20.31985668681424583155669653568, 21.13218062405948605847968981781, 21.73223296748648379623134949424