| L(s) = 1 | + (−0.877 + 0.479i)2-s + (0.997 − 0.0713i)3-s + (0.540 − 0.841i)4-s + (2.06 − 0.867i)5-s + (−0.841 + 0.540i)6-s + (−1.60 + 4.30i)7-s + (−0.0713 + 0.997i)8-s + (0.989 − 0.142i)9-s + (−1.39 + 1.74i)10-s + (1.86 + 6.33i)11-s + (0.479 − 0.877i)12-s + (−2.47 + 0.921i)13-s + (−0.654 − 4.55i)14-s + (1.99 − 1.01i)15-s + (−0.415 − 0.909i)16-s + (−5.68 − 1.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.620 + 0.338i)2-s + (0.575 − 0.0411i)3-s + (0.270 − 0.420i)4-s + (0.921 − 0.388i)5-s + (−0.343 + 0.220i)6-s + (−0.607 + 1.62i)7-s + (−0.0252 + 0.352i)8-s + (0.329 − 0.0474i)9-s + (−0.440 + 0.553i)10-s + (0.561 + 1.91i)11-s + (0.138 − 0.253i)12-s + (−0.685 + 0.255i)13-s + (−0.174 − 1.21i)14-s + (0.514 − 0.261i)15-s + (−0.103 − 0.227i)16-s + (−1.37 − 0.299i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.852175 + 0.993164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.852175 + 0.993164i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.877 - 0.479i)T \) |
| 3 | \( 1 + (-0.997 + 0.0713i)T \) |
| 5 | \( 1 + (-2.06 + 0.867i)T \) |
| 23 | \( 1 + (4.35 - 2.01i)T \) |
| good | 7 | \( 1 + (1.60 - 4.30i)T + (-5.29 - 4.58i)T^{2} \) |
| 11 | \( 1 + (-1.86 - 6.33i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (2.47 - 0.921i)T + (9.82 - 8.51i)T^{2} \) |
| 17 | \( 1 + (5.68 + 1.23i)T + (15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (2.22 + 1.42i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.61 - 2.51i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.46 - 5.15i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-5.71 + 7.63i)T + (-10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (0.533 - 3.71i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (0.667 + 9.33i)T + (-42.5 + 6.11i)T^{2} \) |
| 47 | \( 1 + (0.431 - 0.431i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.32 - 1.24i)T + (40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (-6.21 - 2.83i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-8.79 + 7.62i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-3.40 - 6.23i)T + (-36.2 + 56.3i)T^{2} \) |
| 71 | \( 1 + (-5.72 - 1.68i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.15 - 9.92i)T + (-66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (-0.0599 + 0.131i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.922 - 0.690i)T + (23.3 + 79.6i)T^{2} \) |
| 89 | \( 1 + (-0.183 + 0.212i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-6.96 + 5.21i)T + (27.3 - 93.0i)T^{2} \) |
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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
−10.13375783361037291323199659892, −9.614494980582693014246503473130, −9.061615324124951093066688970602, −8.474084400208496981280742980772, −7.03824912353357163746984028657, −6.53684542141371120309234072963, −5.39513556559076811253797742237, −4.45274070663960503597794604171, −2.35151354265694562473435113056, −2.09438719147966862672227329966,
0.76196807934648902351825650925, 2.36100397208298116848432581937, 3.41117712335434460245191495629, 4.32302619951973800812174147821, 6.26230163339586401362625622953, 6.58053845447284344032258044638, 7.83091184068867870944328278553, 8.561278112028348159436031611246, 9.573688257318110745888921855957, 10.15295501288888147471376471045
