| L(s) = 1 | + (1.91 + 0.561i)2-s + (−0.0122 + 0.0141i)3-s + (1.66 + 1.06i)4-s + (−0.142 + 0.989i)5-s + (−0.0313 + 0.0201i)6-s + (−1.60 − 3.50i)7-s + (−0.0305 − 0.0352i)8-s + (0.426 + 2.96i)9-s + (−0.828 + 1.81i)10-s + (−2.05 + 0.604i)11-s + (−0.0354 + 0.0104i)12-s + (−1.09 + 2.40i)13-s + (−1.09 − 7.61i)14-s + (−0.0122 − 0.0141i)15-s + (−1.68 − 3.68i)16-s + (6.86 − 4.41i)17-s + ⋯ |
| L(s) = 1 | + (1.35 + 0.397i)2-s + (−0.00707 + 0.00816i)3-s + (0.831 + 0.534i)4-s + (−0.0636 + 0.442i)5-s + (−0.0128 + 0.00823i)6-s + (−0.605 − 1.32i)7-s + (−0.0107 − 0.0124i)8-s + (0.142 + 0.989i)9-s + (−0.261 + 0.573i)10-s + (−0.620 + 0.182i)11-s + (−0.0102 + 0.00300i)12-s + (−0.304 + 0.667i)13-s + (−0.292 − 2.03i)14-s + (−0.00316 − 0.00365i)15-s + (−0.420 − 0.920i)16-s + (1.66 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73398 + 0.402620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73398 + 0.402620i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.705 - 4.74i)T \) |
| good | 2 | \( 1 + (-1.91 - 0.561i)T + (1.68 + 1.08i)T^{2} \) |
| 3 | \( 1 + (0.0122 - 0.0141i)T + (-0.426 - 2.96i)T^{2} \) |
| 7 | \( 1 + (1.60 + 3.50i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (2.05 - 0.604i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.09 - 2.40i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-6.86 + 4.41i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (2.95 + 1.89i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (3.45 - 2.22i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.76 - 2.03i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.01 - 7.05i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.0496 - 0.345i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.87 + 4.46i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + (0.962 + 2.10i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.230 - 0.504i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-2.09 - 2.41i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (11.7 + 3.44i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-1.74 - 0.513i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.41 - 1.55i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (4.30 - 9.43i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (2.15 + 14.9i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-3.70 + 4.27i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.24 + 8.67i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−13.72906790015843518271453242356, −13.04531046372508646909896077392, −11.85909728345357638808842701355, −10.60720368099026769106104969749, −9.692652702019405876538095695208, −7.51083696705901676715232030155, −7.01614110063813417966274723391, −5.48055312974462617668104212842, −4.36588025184063217837608958675, −3.06958687469173299346487020481,
2.68816756111252333870916394580, 3.92280984934816565641257454659, 5.56043868552794427714042609530, 6.05993535811594038360158837736, 8.139739086341897662392621483754, 9.291225116050231704450170012246, 10.59741958193523174667333159346, 12.24725731502156718566409876090, 12.33220936793326808593717077403, 13.11018103613604493860967188666
