L(s) = 1 | + 1.93i·5-s + i·7-s + 3.58·11-s + 5.92·13-s + 6.96i·17-s − 5.33i·19-s + 2.96·23-s + 1.26·25-s + 3.06i·29-s − 1.27i·31-s − 1.93·35-s + 9.65·37-s − 8.10i·41-s − 2.46i·43-s + 6.45·47-s + ⋯ |
L(s) = 1 | + 0.863i·5-s + 0.377i·7-s + 1.08·11-s + 1.64·13-s + 1.68i·17-s − 1.22i·19-s + 0.618·23-s + 0.253·25-s + 0.569i·29-s − 0.228i·31-s − 0.326·35-s + 1.58·37-s − 1.26i·41-s − 0.375i·43-s + 0.941·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.609360683\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.609360683\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 1.93iT - 5T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 - 5.92T + 13T^{2} \) |
| 17 | \( 1 - 6.96iT - 17T^{2} \) |
| 19 | \( 1 + 5.33iT - 19T^{2} \) |
| 23 | \( 1 - 2.96T + 23T^{2} \) |
| 29 | \( 1 - 3.06iT - 29T^{2} \) |
| 31 | \( 1 + 1.27iT - 31T^{2} \) |
| 37 | \( 1 - 9.65T + 37T^{2} \) |
| 41 | \( 1 + 8.10iT - 41T^{2} \) |
| 43 | \( 1 + 2.46iT - 43T^{2} \) |
| 47 | \( 1 - 6.45T + 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 6.34T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 6.26iT - 67T^{2} \) |
| 71 | \( 1 + 1.92T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 + 8.71iT - 79T^{2} \) |
| 83 | \( 1 - 7.96T + 83T^{2} \) |
| 89 | \( 1 - 9.04iT - 89T^{2} \) |
| 97 | \( 1 + 4.33T + 97T^{2} \) |
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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
−8.337972586135943968148622668662, −7.29859171775816437975851151377, −6.62673331750784589759227676852, −6.20235067024545578362610945316, −5.50022025291138955002289544575, −4.31748671995699786813381300806, −3.71585464969864007984267814887, −2.99640361491848583706276247906, −1.96190257851591976512129777984, −0.991611024287603634920958340091,
0.961048464826185538801467684908, 1.28717850208480018127886828470, 2.75210415695275946218257571570, 3.67860612169501390090296309438, 4.33185271841652993126674843987, 4.99939876266530047104752732639, 6.01703227946562831903002022026, 6.38207855994970561316016933483, 7.39060731630497799000977884785, 7.984896280173008915384500908863