L(s) = 1 | + (1.11 + 0.864i)2-s + (0.170 + 1.72i)3-s + (0.506 + 1.93i)4-s + (−1.36 − 1.77i)5-s + (−1.29 + 2.07i)6-s + (2.06 − 2.06i)7-s + (−1.10 + 2.60i)8-s + (−2.94 + 0.586i)9-s + (0.00136 − 3.16i)10-s − 0.510·11-s + (−3.24 + 1.20i)12-s + (0.750 − 0.750i)13-s + (4.10 − 0.528i)14-s + (2.81 − 2.65i)15-s + (−3.48 + 1.95i)16-s + (−3.14 − 3.14i)17-s + ⋯ |
L(s) = 1 | + (0.791 + 0.611i)2-s + (0.0982 + 0.995i)3-s + (0.253 + 0.967i)4-s + (−0.610 − 0.791i)5-s + (−0.530 + 0.847i)6-s + (0.782 − 0.782i)7-s + (−0.390 + 0.920i)8-s + (−0.980 + 0.195i)9-s + (0.000431 − 0.999i)10-s − 0.153·11-s + (−0.937 + 0.346i)12-s + (0.208 − 0.208i)13-s + (1.09 − 0.141i)14-s + (0.727 − 0.685i)15-s + (−0.871 + 0.489i)16-s + (−0.763 − 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16267 + 0.950837i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16267 + 0.950837i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.11 - 0.864i)T \) |
| 3 | \( 1 + (-0.170 - 1.72i)T \) |
| 5 | \( 1 + (1.36 + 1.77i)T \) |
good | 7 | \( 1 + (-2.06 + 2.06i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.510T + 11T^{2} \) |
| 13 | \( 1 + (-0.750 + 0.750i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.14 + 3.14i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.01T + 19T^{2} \) |
| 23 | \( 1 + (-2.54 + 2.54i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.10iT - 29T^{2} \) |
| 31 | \( 1 + 4.56T + 31T^{2} \) |
| 37 | \( 1 + (6.76 + 6.76i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 + (5.95 - 5.95i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.33 - 3.33i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.75 + 5.75i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.16iT - 59T^{2} \) |
| 61 | \( 1 + 4.92iT - 61T^{2} \) |
| 67 | \( 1 + (-7.98 - 7.98i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.09iT - 71T^{2} \) |
| 73 | \( 1 + (3.20 + 3.20i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.31iT - 79T^{2} \) |
| 83 | \( 1 + (-4.77 - 4.77i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + (-10.8 + 10.8i)T - 97iT^{2} \) |
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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
−13.94294859174323265625263801187, −12.86725966878911565946392490950, −11.58217855680395537978459905027, −10.93376510061583243033972794924, −9.230785472489144334960243745858, −8.238332449873512320083811218377, −7.22518744718147832966413744948, −5.27721991753884187374161880759, −4.63125321007895621233262274311, −3.43550736402308166291778606064,
2.01028538203947900318104934387, 3.42350360609839962217473166867, 5.26400442406829410998135509842, 6.48813289476535918673099142238, 7.62532109267150214532807456407, 8.925524905768040347093902966566, 10.62297155046823761382486323904, 11.63539300044132453886187739008, 11.97705926480159043151420897692, 13.29070233334773036911228539764