| L(s) = 1 | + 1.07i·2-s + 0.846·4-s − 1.27i·5-s − 1.02i·7-s + 3.05i·8-s + 1.37·10-s + 4.66i·11-s + (1.25 − 3.37i)13-s + 1.10·14-s − 1.58·16-s + 0.476·17-s + 6.69i·19-s − 1.08i·20-s − 5.00·22-s − 0.959·23-s + ⋯ |
| L(s) = 1 | + 0.759i·2-s + 0.423·4-s − 0.570i·5-s − 0.388i·7-s + 1.08i·8-s + 0.433·10-s + 1.40i·11-s + (0.348 − 0.937i)13-s + 0.295·14-s − 0.397·16-s + 0.115·17-s + 1.53i·19-s − 0.241i·20-s − 1.06·22-s − 0.200·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.958336653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958336653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (-1.25 + 3.37i)T \) |
| good | 2 | \( 1 - 1.07iT - 2T^{2} \) |
| 5 | \( 1 + 1.27iT - 5T^{2} \) |
| 7 | \( 1 + 1.02iT - 7T^{2} \) |
| 11 | \( 1 - 4.66iT - 11T^{2} \) |
| 17 | \( 1 - 0.476T + 17T^{2} \) |
| 19 | \( 1 - 6.69iT - 19T^{2} \) |
| 23 | \( 1 + 0.959T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 + 1.92iT - 31T^{2} \) |
| 37 | \( 1 + 4.94iT - 37T^{2} \) |
| 41 | \( 1 - 1.52iT - 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 - 6.83iT - 47T^{2} \) |
| 53 | \( 1 + 0.582T + 53T^{2} \) |
| 59 | \( 1 - 4.21iT - 59T^{2} \) |
| 61 | \( 1 - 9.43T + 61T^{2} \) |
| 67 | \( 1 + 2.32iT - 67T^{2} \) |
| 71 | \( 1 - 1.35iT - 71T^{2} \) |
| 73 | \( 1 + 12.8iT - 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 10.2iT - 83T^{2} \) |
| 89 | \( 1 + 6.85iT - 89T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
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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
−10.23144625519937934231350133518, −9.083897189945730653861177846847, −8.087957682368156720916609004048, −7.66014246148878087822300131300, −6.73873859926857088671858420153, −5.88750810405264662596763229059, −5.04745906605917245560260976232, −4.09304498846320531841564038352, −2.67106902751382342977241826404, −1.36693910259141465796838695912,
1.00732606338612141969411735801, 2.51332945734759766782907117912, 3.09003064952263820762007106425, 4.19252040647058276259269361865, 5.49987063035003408008526810036, 6.67039410585425016186593213619, 6.83692236342143562858638388114, 8.341843761070117300508516082876, 8.954281272890085940682418574293, 9.976079543352781477804496085492
