L(s) = 1 | + (−0.372 + 2.35i)2-s + (0.743 − 1.93i)3-s + (−3.48 − 1.13i)4-s + (2.11 − 0.711i)5-s + (4.27 + 2.47i)6-s + (1.00 + 0.0525i)7-s + (1.80 − 3.53i)8-s + (−0.972 − 0.875i)9-s + (0.882 + 5.24i)10-s + (−0.0960 − 0.451i)11-s + (−4.78 + 5.91i)12-s + (3.31 + 4.09i)13-s + (−0.496 + 2.33i)14-s + (0.198 − 4.63i)15-s + (1.70 + 1.23i)16-s + (−2.01 − 3.10i)17-s + ⋯ |
L(s) = 1 | + (−0.263 + 1.66i)2-s + (0.429 − 1.11i)3-s + (−1.74 − 0.566i)4-s + (0.948 − 0.318i)5-s + (1.74 + 1.00i)6-s + (0.379 + 0.0198i)7-s + (0.636 − 1.24i)8-s + (−0.324 − 0.291i)9-s + (0.279 + 1.65i)10-s + (−0.0289 − 0.136i)11-s + (−1.38 + 1.70i)12-s + (0.920 + 1.13i)13-s + (−0.132 + 0.624i)14-s + (0.0512 − 1.19i)15-s + (0.426 + 0.309i)16-s + (−0.488 − 0.752i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06182 + 0.577240i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06182 + 0.577240i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.11 + 0.711i)T \) |
| 31 | \( 1 + (4.70 - 2.98i)T \) |
good | 2 | \( 1 + (0.372 - 2.35i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.743 + 1.93i)T + (-2.22 - 2.00i)T^{2} \) |
| 7 | \( 1 + (-1.00 - 0.0525i)T + (6.96 + 0.731i)T^{2} \) |
| 11 | \( 1 + (0.0960 + 0.451i)T + (-10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.31 - 4.09i)T + (-2.70 + 12.7i)T^{2} \) |
| 17 | \( 1 + (2.01 + 3.10i)T + (-6.91 + 15.5i)T^{2} \) |
| 19 | \( 1 + (4.30 - 0.452i)T + (18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (3.03 + 1.54i)T + (13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (7.72 - 5.61i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-9.19 + 2.46i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (2.07 + 0.925i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-3.57 - 2.89i)T + (8.94 + 42.0i)T^{2} \) |
| 47 | \( 1 + (8.36 - 1.32i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.338 - 6.45i)T + (-52.7 + 5.54i)T^{2} \) |
| 59 | \( 1 + (4.06 + 9.12i)T + (-39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 2.80iT - 61T^{2} \) |
| 67 | \( 1 + (13.1 + 3.52i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.09 + 2.32i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.624 + 0.961i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-9.59 - 2.03i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-1.29 + 0.496i)T + (61.6 - 55.5i)T^{2} \) |
| 89 | \( 1 + (1.17 - 3.61i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.35 - 10.5i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−13.47634377262334501823457483666, −12.70482402428666630159570129779, −11.06667203005412434479076639223, −9.341861306570114329966060801745, −8.724903371076999529340641080978, −7.75134368337262755128108070822, −6.71665549785106468560290875184, −6.05597853902927709270212412838, −4.68177992930267143761534535015, −1.84421452278607457867884421249,
1.97327679923130994529330080379, 3.36549993007439390339047572803, 4.37318329320738368767085293382, 5.95665749162405577565295750452, 8.236408171296832403701752651308, 9.226681608839605491262001718540, 9.971340007949364942457314234582, 10.67636848412579416100876655109, 11.30326158510906652151883675725, 12.92519339614468881250394147834