| L(s) = 1 | − 2.44·3-s − 2.44i·5-s + (−2.44 + i)7-s + 2.99·9-s + 2i·11-s − 2.44i·13-s + 5.99i·15-s + 4.89i·17-s + 2.44·19-s + (5.99 − 2.44i)21-s + 4i·23-s − 0.999·25-s + 4·29-s − 4.89·31-s − 4.89i·33-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 1.09i·5-s + (−0.925 + 0.377i)7-s + 0.999·9-s + 0.603i·11-s − 0.679i·13-s + 1.54i·15-s + 1.18i·17-s + 0.561·19-s + (1.30 − 0.534i)21-s + 0.834i·23-s − 0.199·25-s + 0.742·29-s − 0.879·31-s − 0.852i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6552515475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6552515475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.44 - i)T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 2.44iT - 5T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 + 7.34iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−9.149098875478999304518557703688, −8.492357220767220337616832429606, −7.37694353876236313957352392611, −6.57780206148932177669843248804, −5.61375500941123443978476593547, −5.39676689209153270887632563842, −4.38150009534384657098008713447, −3.31979739521265682354596216654, −1.67981378046781151188319060897, −0.44586757487140784037279395617,
0.789314099347037547854179069621, 2.64307360221573083493627350321, 3.49681906302410033716260317727, 4.62882904224517515245740995066, 5.55430642434611354518589245045, 6.35957068000839944793288951178, 6.84664746738566026138153225728, 7.39412507812138977735118444724, 8.798059639161744494330478133191, 9.643725280616775914743624024385
