| L(s) = 1 | + (−0.559 + 1.29i)2-s + (1.13 + 1.30i)3-s + (−1.37 − 1.45i)4-s + (0.778 − 2.09i)5-s + (−2.33 + 0.749i)6-s + 3.14·7-s + (2.65 − 0.971i)8-s + (−0.405 + 2.97i)9-s + (2.28 + 2.18i)10-s + (−1.76 − 1.28i)11-s + (0.331 − 3.44i)12-s + (2.63 + 3.62i)13-s + (−1.76 + 4.08i)14-s + (3.62 − 1.37i)15-s + (−0.223 + 3.99i)16-s + (1.48 − 4.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.395 + 0.918i)2-s + (0.657 + 0.753i)3-s + (−0.687 − 0.726i)4-s + (0.348 − 0.937i)5-s + (−0.952 + 0.305i)6-s + 1.19·7-s + (0.939 − 0.343i)8-s + (−0.135 + 0.990i)9-s + (0.723 + 0.690i)10-s + (−0.531 − 0.386i)11-s + (0.0956 − 0.995i)12-s + (0.731 + 1.00i)13-s + (−0.470 + 1.09i)14-s + (0.935 − 0.354i)15-s + (−0.0559 + 0.998i)16-s + (0.360 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15658 + 0.849062i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15658 + 0.849062i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.559 - 1.29i)T \) |
| 3 | \( 1 + (-1.13 - 1.30i)T \) |
| 5 | \( 1 + (-0.778 + 2.09i)T \) |
| good | 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 + (1.76 + 1.28i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.63 - 3.62i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.48 + 4.57i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.98 - 0.968i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (4.21 - 5.80i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.17 + 0.382i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (5.04 + 1.63i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.18 + 7.12i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.03 - 4.18i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 0.651T + 43T^{2} \) |
| 47 | \( 1 + (0.972 - 0.315i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.802 - 2.47i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.99 - 2.17i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 7.51i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.32 + 10.2i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.91 + 12.0i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.85 + 9.44i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (10.8 - 3.53i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.25 + 1.38i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (1.83 - 2.52i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.13 - 1.01i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.73053442426316524045600843535, −10.77216391361946043721297638548, −9.577434291132986510737731326923, −9.085525003202118429449613735527, −8.154730264969322524712198319692, −7.52696551279922422987874146834, −5.67439913536342046090132095763, −5.04418891427075363426323403660, −4.00280358267663196257016622395, −1.68292087622814438897260796461,
1.55348947643128093461258557938, 2.65055369646758110744352603613, 3.80189313408874687703806482921, 5.51404434601416574174146041207, 7.01685926301938902294578307808, 8.078030737903580026239747167444, 8.436578062745898778002691734772, 9.916574199262698537798442936674, 10.61013042883560961576101013218, 11.45112309381651682407223275277
