| L(s) = 1 | + (0.841 + 0.540i)2-s + (0.00806 + 0.0561i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (−0.0235 + 0.0515i)6-s + (−3.27 + 3.78i)7-s + (−0.142 + 0.989i)8-s + (2.87 − 0.844i)9-s + (0.654 + 0.755i)10-s + (1.87 − 1.20i)11-s + (−0.0476 + 0.0306i)12-s + (1.96 + 2.26i)13-s + (−4.80 + 1.41i)14-s + (−0.00806 + 0.0561i)15-s + (−0.654 + 0.755i)16-s + (0.486 − 1.06i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.00465 + 0.0323i)3-s + (0.207 + 0.454i)4-s + (0.429 + 0.125i)5-s + (−0.00961 + 0.0210i)6-s + (−1.23 + 1.43i)7-s + (−0.0503 + 0.349i)8-s + (0.958 − 0.281i)9-s + (0.207 + 0.238i)10-s + (0.564 − 0.362i)11-s + (−0.0137 + 0.00884i)12-s + (0.544 + 0.628i)13-s + (−1.28 + 0.376i)14-s + (−0.00208 + 0.0144i)15-s + (−0.163 + 0.188i)16-s + (0.118 − 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44918 + 0.894815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44918 + 0.894815i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (1.65 + 4.50i)T \) |
| good | 3 | \( 1 + (-0.00806 - 0.0561i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (3.27 - 3.78i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.87 + 1.20i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.96 - 2.26i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.486 + 1.06i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (3.02 + 6.61i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.15 + 2.53i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.12 - 7.82i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.478 - 0.140i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (5.31 + 1.56i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.47 + 10.2i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 4.15T + 47T^{2} \) |
| 53 | \( 1 + (1.78 - 2.06i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.06 - 1.22i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.795 + 5.52i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.27 - 3.39i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-8.24 - 5.30i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.699 - 1.53i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (7.24 + 8.36i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.31 + 0.387i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.30 - 16.0i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.73 - 1.39i)T + (81.6 + 52.4i)T^{2} \) |
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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
−12.50220899401367179999456202218, −11.76924303835438724714366243527, −10.41183199466565003195107052011, −9.216480117586866703422716326339, −8.723300091003875027308750805622, −6.67450295226393910332433442725, −6.54411341934569045079747459619, −5.20144950999534254439488339862, −3.76495900998367337693052249571, −2.43705124015000398475070953267,
1.46885809222305735519007287077, 3.52979753263391086443708327112, 4.25895881189896977380004224894, 5.91855569865691178307081611990, 6.77846473161020940518176195633, 7.86092408809728364918055796114, 9.699524366175827550207333471862, 10.05613823688061803262316965906, 10.93307608899016267069653763921, 12.38609077612519562372491139032
