| L(s) = 1 | + (0.587 − 0.809i)2-s + (2.84 − 0.923i)3-s + (−0.309 − 0.951i)4-s + (−0.152 + 2.23i)5-s + (0.923 − 2.84i)6-s + i·7-s + (−0.951 − 0.309i)8-s + (4.79 − 3.48i)9-s + (1.71 + 1.43i)10-s + (−2.02 − 1.46i)11-s + (−1.75 − 2.41i)12-s + (−0.572 − 0.788i)13-s + (0.809 + 0.587i)14-s + (1.62 + 6.48i)15-s + (−0.809 + 0.587i)16-s + (1.56 + 0.509i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (1.64 − 0.533i)3-s + (−0.154 − 0.475i)4-s + (−0.0682 + 0.997i)5-s + (0.376 − 1.16i)6-s + 0.377i·7-s + (−0.336 − 0.109i)8-s + (1.59 − 1.16i)9-s + (0.542 + 0.453i)10-s + (−0.609 − 0.442i)11-s + (−0.507 − 0.697i)12-s + (−0.158 − 0.218i)13-s + (0.216 + 0.157i)14-s + (0.419 + 1.67i)15-s + (−0.202 + 0.146i)16-s + (0.380 + 0.123i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.27899 - 1.08775i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.27899 - 1.08775i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 + (0.152 - 2.23i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (-2.84 + 0.923i)T + (2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (2.02 + 1.46i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.572 + 0.788i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 0.509i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 4.98i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.90 - 5.37i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 2.35i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.67 - 8.24i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (6.73 + 9.27i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.96 - 5.05i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.97iT - 43T^{2} \) |
| 47 | \( 1 + (-3.71 + 1.20i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.64 + 0.859i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.09 + 5.88i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.99 + 1.44i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-9.00 - 2.92i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (0.311 + 0.959i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.56 + 11.7i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.690 - 2.12i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.45 + 1.12i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (14.0 + 10.2i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.17 + 0.380i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.40777734443100355864365553616, −10.39374637914886753825718235701, −9.510366012646536519955580088076, −8.580062653374886614909563287094, −7.62674417002781852462643082602, −6.79366690762516458534869529575, −5.35655621762122769653329075548, −3.56884592536747180474563431316, −2.99982515070481123521360261513, −1.95801266190515808898824792820,
2.17704064235041209518068228085, 3.68293965639236660770192640216, 4.39421940008948699367763780107, 5.49526396683035177361477750348, 7.18095511750906198614391596670, 8.116610922730353441450576918049, 8.499840188572966820258565215539, 9.684890472172757438453395105260, 10.18773184348613373481775175598, 11.98863011434235334309085402758
