| L(s) = 1 | + (0.479 − 0.877i)2-s + (0.125 − 1.74i)3-s + (−0.540 − 0.841i)4-s + (−0.402 − 2.19i)5-s + (−1.47 − 0.948i)6-s + (0.229 − 0.0857i)7-s + (−0.997 + 0.0713i)8-s + (−0.0744 − 0.0106i)9-s + (−2.12 − 0.700i)10-s + (−1.48 + 5.06i)11-s + (−1.53 + 0.840i)12-s + (−0.499 + 1.33i)13-s + (0.0349 − 0.242i)14-s + (−3.89 + 0.429i)15-s + (−0.415 + 0.909i)16-s + (−1.44 − 6.62i)17-s + ⋯ |
| L(s) = 1 | + (0.338 − 0.620i)2-s + (0.0722 − 1.00i)3-s + (−0.270 − 0.420i)4-s + (−0.180 − 0.983i)5-s + (−0.602 − 0.387i)6-s + (0.0869 − 0.0324i)7-s + (−0.352 + 0.0252i)8-s + (−0.0248 − 0.00356i)9-s + (−0.671 − 0.221i)10-s + (−0.448 + 1.52i)11-s + (−0.444 + 0.242i)12-s + (−0.138 + 0.371i)13-s + (0.00933 − 0.0649i)14-s + (−1.00 + 0.110i)15-s + (−0.103 + 0.227i)16-s + (−0.349 − 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.508138 - 1.28976i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.508138 - 1.28976i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.479 + 0.877i)T \) |
| 5 | \( 1 + (0.402 + 2.19i)T \) |
| 23 | \( 1 + (-4.20 - 2.30i)T \) |
| good | 3 | \( 1 + (-0.125 + 1.74i)T + (-2.96 - 0.426i)T^{2} \) |
| 7 | \( 1 + (-0.229 + 0.0857i)T + (5.29 - 4.58i)T^{2} \) |
| 11 | \( 1 + (1.48 - 5.06i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.499 - 1.33i)T + (-9.82 - 8.51i)T^{2} \) |
| 17 | \( 1 + (1.44 + 6.62i)T + (-15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (-6.43 + 4.13i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (3.62 - 5.64i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.28 + 2.63i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.69 + 1.26i)T + (10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-0.230 - 1.60i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-9.13 - 0.653i)T + (42.5 + 6.11i)T^{2} \) |
| 47 | \( 1 + (1.51 - 1.51i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.284 + 0.762i)T + (-40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (3.27 - 1.49i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.49 - 5.63i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-10.0 - 5.47i)T + (36.2 + 56.3i)T^{2} \) |
| 71 | \( 1 + (4.61 - 1.35i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (9.02 + 1.96i)T + (66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (1.23 + 2.70i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-7.53 - 10.0i)T + (-23.3 + 79.6i)T^{2} \) |
| 89 | \( 1 + (7.51 + 8.67i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.636 - 0.850i)T + (-27.3 - 93.0i)T^{2} \) |
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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
−11.98921757455488398379218736794, −11.32288625228068416094068553770, −9.697062110179706489502362447783, −9.179953007904345457316875921609, −7.56607879201118071676140143927, −7.08749904638092702134211431333, −5.23302734061777311275840886499, −4.52474597058885639661487965959, −2.56333265149216553531405653603, −1.17192964859322024182945749921,
3.15073398107713192640970875909, 3.93430141611201065104060061101, 5.40805059760548020407947843456, 6.29619812678375843298434311955, 7.64087505634329700849986519122, 8.499513720385276933883886681764, 9.788384918414219028958096622011, 10.64262708074663075260195817287, 11.35659314908515561346559476487, 12.72707766094316668600553194211
