| L(s) = 1 | + (−0.755 − 0.654i)2-s + (−1.23 − 1.21i)3-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (0.135 + 1.72i)6-s + (0.0399 + 0.136i)7-s + (0.540 − 0.841i)8-s + (0.0409 + 2.99i)9-s + (0.281 − 0.959i)10-s + (−2.41 − 2.79i)11-s + (1.02 − 1.39i)12-s + (2.41 + 0.707i)13-s + (0.0589 − 0.128i)14-s + (0.594 − 1.62i)15-s + (−0.959 + 0.281i)16-s + (0.338 − 2.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.534 − 0.463i)2-s + (−0.711 − 0.702i)3-s + (0.0711 + 0.494i)4-s + (0.185 + 0.406i)5-s + (0.0552 + 0.704i)6-s + (0.0150 + 0.0514i)7-s + (0.191 − 0.297i)8-s + (0.0136 + 0.999i)9-s + (0.0890 − 0.303i)10-s + (−0.729 − 0.841i)11-s + (0.296 − 0.402i)12-s + (0.668 + 0.196i)13-s + (0.0157 − 0.0344i)14-s + (0.153 − 0.420i)15-s + (−0.239 + 0.0704i)16-s + (0.0820 − 0.570i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0436081 + 0.190166i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0436081 + 0.190166i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.755 + 0.654i)T \) |
| 3 | \( 1 + (1.23 + 1.21i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (2.28 + 4.21i)T \) |
| good | 7 | \( 1 + (-0.0399 - 0.136i)T + (-5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (2.41 + 2.79i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.41 - 0.707i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.338 + 2.35i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (6.49 - 0.933i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (3.82 + 0.550i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (4.87 + 3.13i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.564 - 0.257i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (5.49 - 2.50i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.0168 - 0.0261i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 6.32iT - 47T^{2} \) |
| 53 | \( 1 + (8.44 - 2.47i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (3.12 - 10.6i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (3.28 - 5.11i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (3.90 + 3.38i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (11.2 + 9.78i)T + (10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.26 + 8.76i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (4.50 - 15.3i)T + (-66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.284 + 0.623i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-5.96 + 3.83i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 4.73i)T + (63.5 - 73.3i)T^{2} \) |
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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
−10.37646149945123863288440758332, −9.061433433514513802931952821628, −8.217728304684116563611801151689, −7.44056717372540597801279788462, −6.37745706239034654152894112802, −5.75199260779769170846929544748, −4.35914153890916429159940532651, −2.89609907160178605763811422029, −1.79261871260388750068388292873, −0.12901919614484688279577895089,
1.78096325642006907833945699067, 3.73082686041856064445907653715, 4.79762732645366882824790137080, 5.62310216675108505990585058471, 6.42698951937348836278312782055, 7.48160455212915780532889836246, 8.529993317623594932381287759486, 9.243096419912835073702065580826, 10.19912478086000168938280940820, 10.63590779772979916666902053062
