| 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
−13.48892857793749939420316909406, −11.97724935073088464756852167456, −11.46443753065304609229409007220, −10.38648709788862356382185851151, −8.846862558214489252284103621460, −7.87699679098887083575558907873, −6.96402947741850510735449868461, −5.21539843795396711299561405576, −4.35867395292842845993495595748, −2.95160996790377939743423264063,
1.63895400169781869821855340911, 3.61697489728026623028736186962, 4.80276738030628124787787368750, 6.16203384382464938825589329441, 7.84163121906912740122290087514, 8.231575094361893630832017176569, 9.921905721036659255529107310330, 11.22924315827556115572886674639, 11.87251788578902920140190942472, 12.68032548165232367709469924513
