L(s) = 1 | + (0.216 − 0.976i)2-s + (−0.869 + 0.494i)3-s + (−0.905 − 0.423i)4-s + (−0.255 − 0.966i)5-s + (0.293 + 0.955i)6-s + (0.0596 + 0.998i)7-s + (−0.610 + 0.792i)8-s + (0.511 − 0.859i)9-s + (−0.999 + 0.0397i)10-s + (0.987 + 0.158i)11-s + (0.996 − 0.0794i)12-s + (0.921 − 0.387i)13-s + (0.987 + 0.158i)14-s + (0.700 + 0.714i)15-s + (0.641 + 0.767i)16-s + (−0.827 + 0.561i)17-s + ⋯ |
L(s) = 1 | + (0.216 − 0.976i)2-s + (−0.869 + 0.494i)3-s + (−0.905 − 0.423i)4-s + (−0.255 − 0.966i)5-s + (0.293 + 0.955i)6-s + (0.0596 + 0.998i)7-s + (−0.610 + 0.792i)8-s + (0.511 − 0.859i)9-s + (−0.999 + 0.0397i)10-s + (0.987 + 0.158i)11-s + (0.996 − 0.0794i)12-s + (0.921 − 0.387i)13-s + (0.987 + 0.158i)14-s + (0.700 + 0.714i)15-s + (0.641 + 0.767i)16-s + (−0.827 + 0.561i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4259316500 - 0.7334102827i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4259316500 - 0.7334102827i\) |
\(L(1)\) |
\(\approx\) |
\(0.6908567907 - 0.4413513966i\) |
\(L(1)\) |
\(\approx\) |
\(0.6908567907 - 0.4413513966i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 317 | \( 1 \) |
good | 2 | \( 1 + (0.216 - 0.976i)T \) |
| 3 | \( 1 + (-0.869 + 0.494i)T \) |
| 5 | \( 1 + (-0.255 - 0.966i)T \) |
| 7 | \( 1 + (0.0596 + 0.998i)T \) |
| 11 | \( 1 + (0.987 + 0.158i)T \) |
| 13 | \( 1 + (0.921 - 0.387i)T \) |
| 17 | \( 1 + (-0.827 + 0.561i)T \) |
| 19 | \( 1 + (0.293 - 0.955i)T \) |
| 23 | \( 1 + (-0.405 - 0.914i)T \) |
| 29 | \( 1 + (0.0596 - 0.998i)T \) |
| 31 | \( 1 + (-0.0198 - 0.999i)T \) |
| 37 | \( 1 + (-0.331 + 0.943i)T \) |
| 41 | \( 1 + (-0.476 - 0.878i)T \) |
| 43 | \( 1 + (0.804 - 0.594i)T \) |
| 47 | \( 1 + (-0.0198 - 0.999i)T \) |
| 53 | \( 1 + (0.293 - 0.955i)T \) |
| 59 | \( 1 + (0.804 + 0.594i)T \) |
| 61 | \( 1 + (-0.177 + 0.984i)T \) |
| 67 | \( 1 + (0.848 - 0.528i)T \) |
| 71 | \( 1 + (0.700 - 0.714i)T \) |
| 73 | \( 1 + (0.578 - 0.815i)T \) |
| 79 | \( 1 + (-0.905 + 0.423i)T \) |
| 83 | \( 1 + (-0.936 - 0.350i)T \) |
| 89 | \( 1 + (0.216 - 0.976i)T \) |
| 97 | \( 1 + (-0.331 + 0.943i)T \) |
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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
−25.341613805423697932427396718382, −24.46854646270346252171016422144, −23.46014863081087177809081284171, −23.08647146478817189738148973654, −22.29995928500877573533173262969, −21.49667332916843989570079154704, −19.87149235087941491938021031148, −18.84421198458229443262954617563, −18.00817986409196303900439731788, −17.39516668054272603716376188338, −16.33661230471910654265869880843, −15.78594121232547944094960450626, −14.167888739329339003752753617634, −14.01296072530078695904978804758, −12.76810330336034625060751035546, −11.58396987747237310485180584761, −10.84028093321575993965384244052, −9.617406346195432220922917813161, −8.16597467504587122312322037427, −7.1137035313601616682790088895, −6.68809103838733227833059368270, −5.744433079585273693517821351741, −4.34104191013645454857833447755, −3.51448513613923056539992489355, −1.29671466053142330490940339858,
0.66931258502137455681075431972, 2.02961848153603025066833279838, 3.74435588637680455723831431269, 4.502815531289736667668495078172, 5.485460065202474245545811547028, 6.34662229194870717637869991885, 8.53379789920394328053710172367, 9.05069196986431670052043678803, 10.08661795370865819446443044938, 11.31554413732063604386871578847, 11.81058622647649589969133301760, 12.63100998917303377331059236850, 13.50914786454293444853208991382, 15.08420573681082207648036365552, 15.65200240613868464548031464312, 16.94332764097365772923665898721, 17.6863594119375865212692220304, 18.60925082744271525127147661035, 19.70368084105600087542106566306, 20.5839344727848504711931625264, 21.28465795543481859769681730019, 22.2880842318218220333828279745, 22.639020893422792747450654998323, 23.98678101971132392137431542800, 24.41182582975613623796858094951