| L(s) = 1 | + (−6.34 + 1.69i)2-s + (7.28 − 12.6i)3-s + (23.4 − 13.5i)4-s + (−20.2 − 20.2i)5-s + (−24.7 + 92.4i)6-s + (21.3 + 5.71i)7-s + (−51.5 + 51.5i)8-s + (−65.6 − 113. i)9-s + (163. + 94.1i)10-s + (19.8 + 74.1i)11-s − 394. i·12-s + (162. + 47.5i)13-s − 144.·14-s + (−403. + 108. i)15-s + (22.4 − 38.8i)16-s + (119. − 69.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.58 + 0.424i)2-s + (0.809 − 1.40i)3-s + (1.46 − 0.846i)4-s + (−0.811 − 0.811i)5-s + (−0.687 + 2.56i)6-s + (0.435 + 0.116i)7-s + (−0.805 + 0.805i)8-s + (−0.811 − 1.40i)9-s + (1.63 + 0.941i)10-s + (0.164 + 0.612i)11-s − 2.74i·12-s + (0.959 + 0.281i)13-s − 0.739·14-s + (−1.79 + 0.480i)15-s + (0.0875 − 0.151i)16-s + (0.415 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.515999 - 0.394861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.515999 - 0.394861i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-162. - 47.5i)T \) |
| good | 2 | \( 1 + (6.34 - 1.69i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-7.28 + 12.6i)T + (-40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (20.2 + 20.2i)T + 625iT^{2} \) |
| 7 | \( 1 + (-21.3 - 5.71i)T + (2.07e3 + 1.20e3i)T^{2} \) |
| 11 | \( 1 + (-19.8 - 74.1i)T + (-1.26e4 + 7.32e3i)T^{2} \) |
| 17 | \( 1 + (-119. + 69.2i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-57.8 + 216. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-491. - 283. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (406. - 704. i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (666. + 666. i)T + 9.23e5iT^{2} \) |
| 37 | \( 1 + (-429. - 1.60e3i)T + (-1.62e6 + 9.37e5i)T^{2} \) |
| 41 | \( 1 + (-907. + 243. i)T + (2.44e6 - 1.41e6i)T^{2} \) |
| 43 | \( 1 + (2.11e3 - 1.22e3i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-908. + 908. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 - 2.10e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (3.41e3 + 915. i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-2.81e3 - 4.87e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.27e3 - 878. i)T + (1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + (1.90e3 - 7.10e3i)T + (-2.20e7 - 1.27e7i)T^{2} \) |
| 73 | \( 1 + (-4.19e3 + 4.19e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 3.95e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-1.38e3 - 1.38e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + (-2.39e3 - 8.94e3i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-751. + 2.80e3i)T + (-7.66e7 - 4.42e7i)T^{2} \) |
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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
−18.75265906310253352299097719507, −17.91913057900869387475654006970, −16.52631260156206879712201202685, −15.05652631705565989370039020789, −13.14887097796083186467322134376, −11.57467100614112636247403341821, −9.068248828611418511755082690682, −8.137102561335101132313596733898, −7.08042203083077367208463707342, −1.26774531802544921021089294041,
3.43229384244993914278830270813, 7.87316456296571806668444875909, 9.000177591877198443848855514708, 10.50516188674079500066202966139, 11.21271513330460290898360636987, 14.44536271016927129905802623234, 15.65309206434827154331775512093, 16.70341248035583554417348191274, 18.44714523772809923483751671791, 19.40685336834503350271152234349
