| L(s) = 1 | + (−0.654 − 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.431 − 0.945i)5-s + (−0.142 − 0.989i)6-s + (−0.959 − 0.281i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.997 + 0.292i)10-s + (−3.63 + 4.19i)11-s + (−0.654 + 0.755i)12-s + (−0.875 + 0.257i)13-s + (0.415 + 0.909i)14-s + (0.874 − 0.562i)15-s + (−0.959 − 0.281i)16-s + (−0.279 − 1.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (0.485 + 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.193 − 0.422i)5-s + (−0.0580 − 0.404i)6-s + (−0.362 − 0.106i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.315 + 0.0926i)10-s + (−1.09 + 1.26i)11-s + (−0.189 + 0.218i)12-s + (−0.242 + 0.0712i)13-s + (0.111 + 0.243i)14-s + (0.225 − 0.145i)15-s + (−0.239 − 0.0704i)16-s + (−0.0678 − 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.694853 + 0.599966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.694853 + 0.599966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (1.42 - 4.58i)T \) |
| good | 5 | \( 1 + (-0.431 + 0.945i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (3.63 - 4.19i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.875 - 0.257i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.279 + 1.94i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (1.01 - 7.04i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.423 + 2.94i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (1.58 - 1.01i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (0.186 + 0.407i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.895 + 1.96i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.41 - 3.47i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + (-1.83 - 0.538i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (1.19 - 0.351i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (6.28 - 4.04i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-8.03 - 9.26i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (6.83 + 7.88i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.06 - 7.42i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (4.78 - 1.40i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.503 + 1.10i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (2.68 + 1.72i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.784 + 1.71i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−9.960292325642845721558136904492, −9.632889218519766050733798549966, −8.709084797573131346121445335201, −7.74401854778882891091406007240, −7.25873722975754939795280053279, −5.75729336369695486231521627033, −4.77904222407759729006075986497, −3.81778723994971849280650002166, −2.67615473460964822266252756377, −1.65643836028893103802017942882,
0.44876179426766067135589590898, 2.36305610203578146208742946934, 3.14815981338147099458618497378, 4.66715749261429824067236577336, 5.78217651601733812516549977231, 6.54049907848599365734548432687, 7.30880939904719195452995072084, 8.294238982589660813069905464738, 8.757197624014305451398603069703, 9.690050623403249971721342039068
