L(s) = 1 | + 4.07i·2-s − 8.61·4-s + (10.3 − 4.10i)5-s − 13.3i·7-s − 2.51i·8-s + (16.7 + 42.3i)10-s + 11.5·11-s − 40.0i·13-s + 54.3·14-s − 58.6·16-s − 93.3i·17-s − 75.1·19-s + (−89.5 + 35.3i)20-s + 47.1i·22-s − 142. i·23-s + ⋯ |
L(s) = 1 | + 1.44i·2-s − 1.07·4-s + (0.930 − 0.367i)5-s − 0.720i·7-s − 0.110i·8-s + (0.529 + 1.34i)10-s + 0.316·11-s − 0.855i·13-s + 1.03·14-s − 0.917·16-s − 1.33i·17-s − 0.906·19-s + (−1.00 + 0.395i)20-s + 0.456i·22-s − 1.28i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.013514977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013514977\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-10.3 + 4.10i)T \) |
good | 2 | \( 1 - 4.07iT - 8T^{2} \) |
| 7 | \( 1 + 13.3iT - 343T^{2} \) |
| 11 | \( 1 - 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 93.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 75.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 82.9iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 449.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 279. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 33.8iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 423. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 615.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 502.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 57.7iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 252.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 823. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 205.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 646. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 563. iT - 9.12e5T^{2} \) |
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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
−10.57875672749137320431993237265, −9.849188362751582570187259979854, −8.717623239697101888489161396567, −8.103296588801661488865380584026, −6.85477779992682386777688124879, −6.39060317900082445638487019942, −5.22095859008939632510251937434, −4.49990351979487754232629971581, −2.58538821611311252746649248808, −0.68987915250111820384085994321,
1.48158260599489355668283431170, 2.24918172593928220130856803872, 3.39582703377686005898217044672, 4.59998284582592909130060955410, 6.00249100183038245968007484003, 6.77089230939430179683330172136, 8.521355437644329656150775009420, 9.225657661096690121343695911853, 10.15106913443489551487567569710, 10.65831213434930570703812377314