| L(s) = 1 | + (0.978 + 0.207i)2-s + (−1.53 − 2.66i)3-s + (0.913 + 0.406i)4-s + (0.318 + 3.02i)5-s + (−0.951 − 2.92i)6-s + (2.44 + 1.01i)7-s + (0.809 + 0.587i)8-s + (−3.23 + 5.60i)9-s + (−0.318 + 3.02i)10-s + (0.257 − 2.45i)11-s + (−0.321 − 3.06i)12-s + (1.44 + 4.45i)13-s + (2.17 + 1.50i)14-s + (7.57 − 5.50i)15-s + (0.669 + 0.743i)16-s + (−0.482 + 4.59i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (−0.888 − 1.53i)3-s + (0.456 + 0.203i)4-s + (0.142 + 1.35i)5-s + (−0.388 − 1.19i)6-s + (0.923 + 0.384i)7-s + (0.286 + 0.207i)8-s + (−1.07 + 1.86i)9-s + (−0.100 + 0.957i)10-s + (0.0776 − 0.738i)11-s + (−0.0928 − 0.883i)12-s + (0.401 + 1.23i)13-s + (0.582 + 0.401i)14-s + (1.95 − 1.42i)15-s + (0.167 + 0.185i)16-s + (−0.117 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81865 + 0.207718i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81865 + 0.207718i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 7 | \( 1 + (-2.44 - 1.01i)T \) |
| 41 | \( 1 + (6.33 - 0.929i)T \) |
| good | 3 | \( 1 + (1.53 + 2.66i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.318 - 3.02i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.257 + 2.45i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.44 - 4.45i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.482 - 4.59i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.24 - 1.38i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.154 + 0.0328i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-6.72 + 4.88i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.865 + 8.23i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.616 + 5.86i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (0.201 + 0.618i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (1.74 + 0.371i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-11.9 - 5.29i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (8.37 - 9.29i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.670 - 0.744i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (0.104 + 0.0464i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (7.72 + 5.61i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.302 + 0.523i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.48 - 9.50i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.56T + 83T^{2} \) |
| 89 | \( 1 + (7.96 + 8.84i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-10.2 + 7.44i)T + (29.9 - 92.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.16668695321059741718441008650, −10.43498023474368592410574121879, −8.578314885297669248377240520687, −7.72997570774434223051331850025, −6.91644821520336210994410572974, −6.14339680806972879078771046115, −5.75609172918818121137517446583, −4.20485363355236800375647130882, −2.59551177267816936324123010480, −1.64467283629047309774712420662,
1.05387627248859537536866428090, 3.28380445906371957733458001015, 4.54536404569271007718132367269, 4.98885234147813164979436606083, 5.38049902452308336593597378693, 6.81471906039555345372323737430, 8.277193062319149425391319364522, 9.119127675253295913840471927306, 10.16131579247832966742275880928, 10.59813895297688997879803994369
