| L(s) = 1 | + (0.593 − 0.804i)2-s + (−1.64 + 0.311i)3-s + (−0.294 − 0.955i)4-s + (1.39 − 1.75i)5-s + (−0.728 + 1.51i)6-s + (2.64 − 0.172i)7-s + (−0.943 − 0.330i)8-s + (−0.171 + 0.0672i)9-s + (−0.582 − 2.15i)10-s + (0.483 − 1.23i)11-s + (0.784 + 1.48i)12-s + (4.81 + 0.542i)13-s + (1.42 − 2.22i)14-s + (−1.74 + 3.32i)15-s + (−0.826 + 0.563i)16-s + (−6.93 − 0.259i)17-s + ⋯ |
| L(s) = 1 | + (0.419 − 0.568i)2-s + (−0.951 + 0.180i)3-s + (−0.147 − 0.477i)4-s + (0.622 − 0.782i)5-s + (−0.297 + 0.617i)6-s + (0.997 − 0.0650i)7-s + (−0.333 − 0.116i)8-s + (−0.0570 + 0.0224i)9-s + (−0.184 − 0.682i)10-s + (0.145 − 0.371i)11-s + (0.226 + 0.428i)12-s + (1.33 + 0.150i)13-s + (0.382 − 0.595i)14-s + (−0.451 + 0.857i)15-s + (−0.206 + 0.140i)16-s + (−1.68 − 0.0628i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.930935 - 1.11697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930935 - 1.11697i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.593 + 0.804i)T \) |
| 5 | \( 1 + (-1.39 + 1.75i)T \) |
| 7 | \( 1 + (-2.64 + 0.172i)T \) |
| good | 3 | \( 1 + (1.64 - 0.311i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-0.483 + 1.23i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-4.81 - 0.542i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (6.93 + 0.259i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-1.16 + 2.00i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.293 + 7.83i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (4.04 - 0.922i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-7.60 + 4.39i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.18 - 2.21i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-4.78 - 9.94i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.93 + 5.51i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (4.14 + 3.05i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-0.584 - 0.309i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.771 - 10.2i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (2.14 - 6.96i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.988 - 0.264i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (2.83 - 12.4i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.99 + 6.64i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (3.19 + 1.84i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.628 - 5.57i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-3.46 - 8.83i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-3.17 - 3.17i)T + 97iT^{2} \) |
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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
−11.04144771064880877046220423639, −10.16560870181081091952683173043, −8.815270500967085286487095185654, −8.450532641139541573431915814416, −6.53232673518039900381593275387, −5.87405855613041936787730028764, −4.84685744731821126938757369130, −4.29712105762116922989452722529, −2.35182629863897133251027289203, −0.932695844075486200538789775006,
1.79478834895333056973549136233, 3.51315891381702060852234371844, 4.85441239717044902370380157815, 5.76915927842134020969467552035, 6.38806505026408059227520995362, 7.26701044292190268832941004674, 8.380521163464015091191168677926, 9.352814698938275528348238892369, 10.74371503393059529826715193266, 11.18976529569235205428216922372
