| L(s) = 1 | + (0.380 + 1.17i)2-s + (−2.02 + 0.431i)3-s + (0.389 − 0.283i)4-s + (0.772 − 1.33i)5-s + (−1.27 − 2.21i)6-s + (−3.47 − 1.54i)7-s + (2.47 + 1.79i)8-s + (1.19 − 0.531i)9-s + (1.86 + 0.395i)10-s + (0.393 + 3.74i)11-s + (−0.668 + 0.742i)12-s + (−1.76 − 1.95i)13-s + (0.489 − 4.66i)14-s + (−0.991 + 3.05i)15-s + (−0.866 + 2.66i)16-s + (−0.394 + 3.75i)17-s + ⋯ |
| L(s) = 1 | + (0.269 + 0.828i)2-s + (−1.17 + 0.249i)3-s + (0.194 − 0.141i)4-s + (0.345 − 0.598i)5-s + (−0.521 − 0.903i)6-s + (−1.31 − 0.584i)7-s + (0.874 + 0.635i)8-s + (0.397 − 0.177i)9-s + (0.589 + 0.125i)10-s + (0.118 + 1.12i)11-s + (−0.193 + 0.214i)12-s + (−0.489 − 0.543i)13-s + (0.130 − 1.24i)14-s + (−0.255 + 0.787i)15-s + (−0.216 + 0.666i)16-s + (−0.0956 + 0.910i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.591864 + 0.250216i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.591864 + 0.250216i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (3.88 - 3.98i)T \) |
| good | 2 | \( 1 + (-0.380 - 1.17i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (2.02 - 0.431i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.772 + 1.33i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (3.47 + 1.54i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.393 - 3.74i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.76 + 1.95i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.394 - 3.75i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-4.08 + 4.53i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.736 + 0.534i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.10 + 6.47i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.907 - 1.57i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.329 + 0.0700i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.59 + 2.88i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (0.367 - 1.13i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.14 - 0.953i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (7.60 - 1.61i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 + (-3.71 + 6.42i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.65 - 2.07i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (0.563 + 5.36i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-1.01 + 9.68i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-8.21 - 1.74i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (4.12 - 2.99i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (8.82 - 6.41i)T + (29.9 - 92.2i)T^{2} \) |
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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
−16.97513268454174128175482095678, −16.10817007265335702924587462823, −15.14606788968162121076561563565, −13.46280752281583176758270480092, −12.35524577570050580398960861080, −10.76293350628672922727970560184, −9.674780228125041538017402437501, −7.25164698547656826192155015842, −6.08672357786122193649285099056, −4.87636712298002155565865665085,
3.10508788532872281543473427084, 5.82483504900735932203209099317, 6.94890037032596182305044928658, 9.600657510813213688221874409660, 10.91609395010105583390893302033, 11.84339625733855223979571121151, 12.70401889904057804493190581589, 14.06624895282221810356105092186, 16.19607750501165294556985677550, 16.56052478539370329345981975182
