| L(s) = 1 | + (−0.415 − 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (2.43 + 1.56i)5-s + (0.654 + 0.755i)6-s + (0.394 + 2.74i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (0.411 − 2.86i)10-s + (2.26 − 4.95i)11-s + (0.415 − 0.909i)12-s + (−0.0520 + 0.361i)13-s + (2.33 − 1.49i)14-s + (−2.77 − 0.815i)15-s + (−0.142 − 0.989i)16-s + (4.12 + 4.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (1.08 + 0.699i)5-s + (0.267 + 0.308i)6-s + (0.149 + 1.03i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.130 − 0.905i)10-s + (0.682 − 1.49i)11-s + (0.119 − 0.262i)12-s + (−0.0144 + 0.100i)13-s + (0.622 − 0.400i)14-s + (−0.716 − 0.210i)15-s + (−0.0355 − 0.247i)16-s + (1.00 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.943931 + 0.0164483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.943931 + 0.0164483i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (0.965 + 4.69i)T \) |
| good | 5 | \( 1 + (-2.43 - 1.56i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.394 - 2.74i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.26 + 4.95i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.0520 - 0.361i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.12 - 4.75i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (4.01 - 4.63i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (2.23 + 2.57i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (7.37 + 2.16i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.06 + 2.61i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (3.56 + 2.29i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.75 + 1.98i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + (1.01 + 7.06i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.0626 - 0.435i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (7.80 + 2.29i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.84 - 4.04i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (0.586 + 1.28i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.58 + 2.97i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.565 + 3.93i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-13.5 + 8.70i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (13.2 - 3.87i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (3.64 + 2.34i)T + (40.2 + 88.2i)T^{2} \) |
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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
−12.94639085085242238334749969267, −12.05868275639747913331384504287, −11.00658400812590545978678489518, −10.31317361972982274125412002501, −9.235760121773541026273125859600, −8.228967587873290978809285033067, −6.19856333974403990951293464482, −5.75851768230333562042675882712, −3.65935103232670695353050439161, −2.01235839338671016318229200224,
1.44568029499968884435664446617, 4.50089422341723099114902884208, 5.43953355367735571031610232312, 6.79914770647445511989703762645, 7.55043159816356926677255863212, 9.296438450697313142122560257042, 9.778987787933814135862995444552, 10.98397770023013197635222334468, 12.38252467415475388395065921284, 13.27368763559427215478645651759
