| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−2.05 − 0.870i)5-s + (−0.309 − 0.951i)6-s + 2.92·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−2.17 + 0.506i)10-s + (−0.154 + 0.111i)11-s + (−0.809 − 0.587i)12-s + (−0.250 − 0.182i)13-s + (2.36 − 1.72i)14-s + (−1.46 + 1.69i)15-s + (−0.809 − 0.587i)16-s + (1.86 + 5.75i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.921 − 0.389i)5-s + (−0.126 − 0.388i)6-s + 1.10·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.688 + 0.160i)10-s + (−0.0464 + 0.0337i)11-s + (−0.233 − 0.169i)12-s + (−0.0695 − 0.0505i)13-s + (0.633 − 0.459i)14-s + (−0.378 + 0.436i)15-s + (−0.202 − 0.146i)16-s + (0.453 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16828 - 0.889980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16828 - 0.889980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (2.05 + 0.870i)T \) |
| good | 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 + (0.154 - 0.111i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.250 + 0.182i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.86 - 5.75i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1 - 3.07i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.61 - 2.62i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.10 + 6.47i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.522 + 1.60i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.89 - 5.73i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.10 + 2.98i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + (3.31 - 10.1i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.08 + 9.48i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (6.00 + 4.36i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.48 + 3.25i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.66 + 14.3i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.11 + 6.51i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (9.41 - 6.84i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.485 + 1.49i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.02 - 12.3i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.51 + 5.45i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.537 + 1.65i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.68032548165232367709469924513, −11.87251788578902920140190942472, −11.22924315827556115572886674639, −9.921905721036659255529107310330, −8.231575094361893630832017176569, −7.84163121906912740122290087514, −6.16203384382464938825589329441, −4.80276738030628124787787368750, −3.61697489728026623028736186962, −1.63895400169781869821855340911,
2.95160996790377939743423264063, 4.35867395292842845993495595748, 5.21539843795396711299561405576, 6.96402947741850510735449868461, 7.87699679098887083575558907873, 8.846862558214489252284103621460, 10.38648709788862356382185851151, 11.46443753065304609229409007220, 11.97724935073088464756852167456, 13.48892857793749939420316909406
