| L(s) = 1 | + (0.0847 − 0.0248i)2-s + (1.89 + 2.18i)3-s + (−1.67 + 1.07i)4-s + (−0.142 − 0.989i)5-s + (0.214 + 0.138i)6-s + (0.565 − 1.23i)7-s + (−0.230 + 0.266i)8-s + (−0.763 + 5.30i)9-s + (−0.0366 − 0.0803i)10-s + (0.569 + 0.167i)11-s + (−5.52 − 1.62i)12-s + (−1.40 − 3.08i)13-s + (0.0170 − 0.118i)14-s + (1.89 − 2.18i)15-s + (1.64 − 3.59i)16-s + (3.05 + 1.96i)17-s + ⋯ |
| L(s) = 1 | + (0.0599 − 0.0175i)2-s + (1.09 + 1.26i)3-s + (−0.837 + 0.538i)4-s + (−0.0636 − 0.442i)5-s + (0.0876 + 0.0563i)6-s + (0.213 − 0.467i)7-s + (−0.0816 + 0.0941i)8-s + (−0.254 + 1.76i)9-s + (−0.0115 − 0.0253i)10-s + (0.171 + 0.0503i)11-s + (−1.59 − 0.468i)12-s + (−0.390 − 0.855i)13-s + (0.00457 − 0.0317i)14-s + (0.488 − 0.564i)15-s + (0.410 − 0.899i)16-s + (0.740 + 0.475i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07819 + 0.620231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07819 + 0.620231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.68 + 1.03i)T \) |
| good | 2 | \( 1 + (-0.0847 + 0.0248i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (-1.89 - 2.18i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-0.565 + 1.23i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.569 - 0.167i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (1.40 + 3.08i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.05 - 1.96i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.90 + 2.51i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (6.83 + 4.39i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.27 + 2.62i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.57 - 10.9i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.893 - 6.21i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (5.63 + 6.50i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + (1.43 - 3.13i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.36 - 7.35i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-9.80 + 11.3i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.36 + 0.399i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (1.96 - 0.577i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (2.19 - 1.40i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.69 - 3.72i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.802 - 5.58i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (6.82 + 7.87i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.84 + 12.8i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−13.82067419736058526206217477330, −13.04899885725675739718073313456, −11.69380494820826843060917878345, −10.05065188072834721041772791486, −9.626436483300285136627907976339, −8.367398042399062117108451663260, −7.82498806452730525451952456832, −5.20794837858386422295312066389, −4.20160391329732884323398108598, −3.19638501862491116084141897788,
1.81273163322882876612072478856, 3.56214013494217359703749072215, 5.52830268111951928327308000455, 6.94408871276851695600831860213, 7.973115330016271436291982210292, 9.018597763910697176497510582647, 9.867546515222479662366309667266, 11.65025335133306916414610012723, 12.58723815921137612112585872325, 13.62500662745961198440057221165
