| L(s) = 1 | + (−1.65 − 0.5i)3-s + 2·4-s + (2.5 + 1.65i)9-s − 3.31i·11-s + (−3.31 − i)12-s + 4·16-s + 3.31·23-s + (−3.31 − 4i)27-s + 5·31-s + (−1.65 + 5.5i)33-s + (5 + 3.31i)36-s − 7i·37-s − 6.63i·44-s + 6.63·47-s + (−6.63 − 2i)48-s − 7·49-s + ⋯ |
| L(s) = 1 | + (−0.957 − 0.288i)3-s + 4-s + (0.833 + 0.552i)9-s − 1.00i·11-s + (−0.957 − 0.288i)12-s + 16-s + 0.691·23-s + (−0.638 − 0.769i)27-s + 0.898·31-s + (−0.288 + 0.957i)33-s + (0.833 + 0.552i)36-s − 1.15i·37-s − 1.00i·44-s + 0.967·47-s + (−0.957 − 0.288i)48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36411 - 0.542069i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36411 - 0.542069i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 + 0.5i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 3.31iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6.63T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 13iT - 67T^{2} \) |
| 71 | \( 1 + 16.5iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−10.59817953643754018459710579546, −9.401067549598391957900840583736, −8.201244368567474408230362158153, −7.38401638004813304291565101335, −6.54209031018467767441435618207, −5.90661780902831983908195123913, −5.01943509722201261642347932023, −3.61014493610218366519615595011, −2.32680577802956869387270719415, −0.931224577032168824816211090960,
1.31573553680812940050952492119, 2.70287172405619777725703801865, 4.09015205073976001720346171378, 5.08771204431907769147813070615, 6.00667128040143966406850595805, 6.87783337062924864247650477087, 7.39196358743748607893498500096, 8.652696098264820913581291741453, 9.965804407657049928176871508267, 10.22507616310788498399521166841
