L(s) = 1 | − 2.41i·2-s + i·3-s − 3.82·4-s + 2.41·6-s + 0.828i·7-s + 4.41i·8-s − 9-s − 11-s − 3.82i·12-s + 5.65i·13-s + 1.99·14-s + 2.99·16-s − 1.17i·17-s + 2.41i·18-s + 6.82·19-s + ⋯ |
L(s) = 1 | − 1.70i·2-s + 0.577i·3-s − 1.91·4-s + 0.985·6-s + 0.313i·7-s + 1.56i·8-s − 0.333·9-s − 0.301·11-s − 1.10i·12-s + 1.56i·13-s + 0.534·14-s + 0.749·16-s − 0.284i·17-s + 0.569i·18-s + 1.56·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17670 - 0.277781i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17670 - 0.277781i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.41iT - 2T^{2} \) |
| 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 1.17iT - 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 8.82iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 9.31iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−10.16548877414478517720417166227, −9.507762931453411854010150708563, −9.072343170812401081747955661232, −7.951175351512570473982404454002, −6.61953017175350998789262794380, −5.22276486110793439709076503918, −4.51050717951824902994725217071, −3.48289237396234181886523872791, −2.62869469164556863918922539673, −1.35448624701722751053163654461,
0.67388100516465070142553089686, 2.86614929481150183736850067960, 4.28667896326212264251382151474, 5.51654273147245108412606694146, 5.82353290354193754799186635818, 7.08290136686817891922072277247, 7.51734065113180192976087766273, 8.245569449669501105313673248466, 9.022766228414750512392367202088, 10.10529241128575786041454706664