L(s) = 1 | − 36.0·2-s − 119. i·3-s + 785.·4-s + 917. i·5-s + 4.31e3i·6-s + 4.19e3i·7-s − 9.83e3·8-s + 5.29e3·9-s − 3.30e4i·10-s − 5.60e4i·11-s − 9.41e4i·12-s + 1.94e3·13-s − 1.51e5i·14-s + 1.10e5·15-s − 4.77e4·16-s + (1.36e4 − 3.44e5i)17-s + ⋯ |
L(s) = 1 | − 1.59·2-s − 0.854i·3-s + 1.53·4-s + 0.656i·5-s + 1.36i·6-s + 0.660i·7-s − 0.849·8-s + 0.269·9-s − 1.04i·10-s − 1.15i·11-s − 1.31i·12-s + 0.0188·13-s − 1.05i·14-s + 0.561·15-s − 0.181·16-s + (0.0396 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0396 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0396 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.455092 - 0.473489i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.455092 - 0.473489i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-1.36e4 + 3.44e5i)T \) |
good | 2 | \( 1 + 36.0T + 512T^{2} \) |
| 3 | \( 1 + 119. iT - 1.96e4T^{2} \) |
| 5 | \( 1 - 917. iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 4.19e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 5.60e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.94e3T + 1.06e10T^{2} \) |
| 19 | \( 1 + 3.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.87e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 3.74e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 3.22e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 7.92e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 2.96e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 1.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.94e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.12e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.42e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.77e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 8.42e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.54e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + 2.56e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 3.39e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 - 5.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.33e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.00e9iT - 7.60e17T^{2} \) |
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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
−16.77749361537817156053449949703, −15.48477185383296527872471906195, −13.64192780567591930123926618320, −11.84326812816295253468522694512, −10.56226295141135683643670229909, −8.998853366636255413905343810494, −7.69620147469517769821876965498, −6.42809019091976398040049055926, −2.37608635176609514704847855056, −0.60389583737269171306815274619,
1.39153181930350342557709442193, 4.44070299484765887180454511503, 7.17943107204821916182933941587, 8.725178007625868243995133898232, 9.906885629583010355067894633430, 10.73977799147965137468144529917, 12.73735519942449449748869917216, 15.00028817918793086183958062724, 16.26050028726272398561975173203, 17.02442895781229302313704903546