| L(s) = 1 | + (1.72 − 0.559i)2-s + (−0.230 + 1.71i)3-s + (1.03 − 0.754i)4-s + (−0.322 + 0.186i)5-s + (0.564 + 3.08i)6-s + (−0.413 − 3.93i)7-s + (−0.763 + 1.05i)8-s + (−2.89 − 0.790i)9-s + (−0.452 + 0.502i)10-s + (2.21 + 0.985i)11-s + (1.05 + 1.95i)12-s + (−0.697 − 3.28i)13-s + (−2.91 − 6.54i)14-s + (−0.245 − 0.597i)15-s + (−1.52 + 4.67i)16-s + (−1.61 + 0.717i)17-s + ⋯ |
| L(s) = 1 | + (1.21 − 0.395i)2-s + (−0.132 + 0.991i)3-s + (0.519 − 0.377i)4-s + (−0.144 + 0.0833i)5-s + (0.230 + 1.26i)6-s + (−0.156 − 1.48i)7-s + (−0.269 + 0.371i)8-s + (−0.964 − 0.263i)9-s + (−0.142 + 0.158i)10-s + (0.667 + 0.297i)11-s + (0.304 + 0.564i)12-s + (−0.193 − 0.909i)13-s + (−0.779 − 1.75i)14-s + (−0.0634 − 0.154i)15-s + (−0.380 + 1.16i)16-s + (−0.390 + 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51510 + 0.100944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51510 + 0.100944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.230 - 1.71i)T \) |
| 31 | \( 1 + (-5.55 + 0.428i)T \) |
| good | 2 | \( 1 + (-1.72 + 0.559i)T + (1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (0.322 - 0.186i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.413 + 3.93i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-2.21 - 0.985i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.697 + 3.28i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (1.61 - 0.717i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (1.48 + 0.315i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.13 - 1.54i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.95 - 9.10i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.46 - 0.844i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.96 + 8.07i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.05 + 9.68i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-2.45 - 0.798i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.371 + 3.53i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-3.35 + 3.01i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 1.12iT - 61T^{2} \) |
| 67 | \( 1 + (-4.76 - 8.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.11 + 0.747i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (4.10 - 9.22i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (2.14 + 4.81i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-6.70 + 7.44i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (10.8 - 7.88i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-7.40 + 5.38i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.03045161056674993539737766496, −13.16398435412609490195974022496, −11.98099508701546147795135434972, −10.91395449466123937587188201584, −10.17531142718481750932116793717, −8.684615066716457741632897820274, −6.92728389614012444014513259864, −5.33394467953261277175024638679, −4.20446833823178141233501979957, −3.35339788673914370566372588045,
2.62380521227688121007719995599, 4.59796311592557286569727265933, 6.05896201432358002965436294573, 6.55308155197829891034265883803, 8.283810098264761798874822473648, 9.397456338831929721900837612630, 11.73702777673900720153791797562, 11.96312537942753835028299852413, 13.07081845107441167069203747341, 13.93516367295374778430867706453
