L(s) = 1 | + (−0.909 − 0.415i)3-s + (−0.142 + 0.989i)5-s + (−0.989 − 0.142i)7-s + (0.654 + 0.755i)9-s + (0.142 + 0.989i)11-s + (0.909 + 0.415i)13-s + (0.540 − 0.841i)15-s + (−0.841 + 0.540i)17-s + (0.755 − 0.654i)19-s + (0.841 + 0.540i)21-s + (0.755 − 0.654i)23-s + (−0.959 − 0.281i)25-s + (−0.281 − 0.959i)27-s + (−0.989 − 0.142i)29-s + (0.755 + 0.654i)31-s + ⋯ |
L(s) = 1 | + (−0.909 − 0.415i)3-s + (−0.142 + 0.989i)5-s + (−0.989 − 0.142i)7-s + (0.654 + 0.755i)9-s + (0.142 + 0.989i)11-s + (0.909 + 0.415i)13-s + (0.540 − 0.841i)15-s + (−0.841 + 0.540i)17-s + (0.755 − 0.654i)19-s + (0.841 + 0.540i)21-s + (0.755 − 0.654i)23-s + (−0.959 − 0.281i)25-s + (−0.281 − 0.959i)27-s + (−0.989 − 0.142i)29-s + (0.755 + 0.654i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1321993815 + 0.4439597249i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1321993815 + 0.4439597249i\) |
\(L(1)\) |
\(\approx\) |
\(0.6199557966 + 0.1563919521i\) |
\(L(1)\) |
\(\approx\) |
\(0.6199557966 + 0.1563919521i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 5 | \( 1 + (-0.142 + 0.989i)T \) |
| 7 | \( 1 + (-0.989 - 0.142i)T \) |
| 11 | \( 1 + (0.142 + 0.989i)T \) |
| 13 | \( 1 + (0.909 + 0.415i)T \) |
| 17 | \( 1 + (-0.841 + 0.540i)T \) |
| 19 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (0.755 - 0.654i)T \) |
| 29 | \( 1 + (-0.989 - 0.142i)T \) |
| 31 | \( 1 + (0.755 + 0.654i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.909 + 0.415i)T \) |
| 43 | \( 1 + (-0.989 + 0.142i)T \) |
| 47 | \( 1 + (-0.415 - 0.909i)T \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.909 + 0.415i)T \) |
| 61 | \( 1 + (0.281 + 0.959i)T \) |
| 67 | \( 1 + (-0.415 + 0.909i)T \) |
| 71 | \( 1 + (0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.654 + 0.755i)T \) |
| 79 | \( 1 + (0.654 - 0.755i)T \) |
| 83 | \( 1 + (-0.540 - 0.841i)T \) |
| 97 | \( 1 + (-0.142 + 0.989i)T \) |
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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
−22.40491374323307743726487663788, −21.483535096217042067324086785456, −20.73160700089740492434122721133, −19.96934886914077706329737027765, −18.91600999219715990428938174123, −18.193519721507609620354930813839, −17.08446299192226633651116201, −16.529008176270750674133074008509, −15.83497117050348337632860964915, −15.36485226810219219407815370590, −13.65801132164830060209341455768, −13.14210840294337005245110423291, −12.21997584924795359926396104192, −11.44209499208324154333815371008, −10.6741835677246564157649527991, −9.46168235242802224382841647350, −9.07857646783633508797252156551, −7.86856849204030185346831607188, −6.600377198971914958451423976085, −5.803428249750622770510025484229, −5.16689086471081033915532917235, −3.9438497407785746939482262171, −3.250924189261550806614933650498, −1.35199023036422018236486784395, −0.27436912101501187830714809109,
1.45196168046861910336295726116, 2.66069758680338994999753223447, 3.78277439212362409777370259790, 4.814471536729411783689583908, 6.07909564321981634919718804932, 6.8293790723377158320281428560, 7.08982241418842051532273255862, 8.53094502018286500816008354633, 9.80194331230411057543384034248, 10.42804110390146461321589460413, 11.3382858449165788718901022114, 11.9652223989943131735128102287, 13.1680033831063429293573260720, 13.475896802427436953013195090596, 14.9010684868099577125554078462, 15.58532570305262668847151208572, 16.43379964754689755052524066163, 17.31897381270596847529955485639, 18.12184069170574978448286069988, 18.728050680750127262683867285493, 19.49462390063511664135572333556, 20.37042968118915201987662942710, 21.6962737691369922002661398663, 22.28359932496457818991610085757, 22.97505677175327085739114172961