| L(s) = 1 | + 2.85·5-s − i·7-s − 4.98i·11-s + 2.35i·13-s − 8.19i·17-s − 5.90·19-s − 4.43·23-s + 3.17·25-s + 6.41·29-s + 3.31i·31-s − 2.85i·35-s − 5.51i·37-s + 3.41i·41-s − 1.37·43-s − 2.60·47-s + ⋯ |
| L(s) = 1 | + 1.27·5-s − 0.377i·7-s − 1.50i·11-s + 0.652i·13-s − 1.98i·17-s − 1.35·19-s − 0.925·23-s + 0.634·25-s + 1.19·29-s + 0.595i·31-s − 0.483i·35-s − 0.905i·37-s + 0.533i·41-s − 0.209·43-s − 0.379·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.485652202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485652202\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 + 4.98iT - 11T^{2} \) |
| 13 | \( 1 - 2.35iT - 13T^{2} \) |
| 17 | \( 1 + 8.19iT - 17T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 - 6.41T + 29T^{2} \) |
| 31 | \( 1 - 3.31iT - 31T^{2} \) |
| 37 | \( 1 + 5.51iT - 37T^{2} \) |
| 41 | \( 1 - 3.41iT - 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 + 1.41iT - 61T^{2} \) |
| 67 | \( 1 + 0.221T + 67T^{2} \) |
| 71 | \( 1 - 0.398T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 9.76iT - 79T^{2} \) |
| 83 | \( 1 - 2.06iT - 83T^{2} \) |
| 89 | \( 1 - 17.6iT - 89T^{2} \) |
| 97 | \( 1 + 3.62T + 97T^{2} \) |
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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
−7.940781680341047656036190245348, −6.75202335625451218822539458487, −6.49993189557918307117348585817, −5.70867005084466585632533718999, −5.00677847084347941956467612541, −4.21440025928716528601000664707, −3.14556371871406141664999885555, −2.44935428649441880814252009342, −1.49101096630246739868047129420, −0.33034567612266357435911965580,
1.64374529198215036106400193049, 1.97981251662471148421056396211, 2.91899727192876284503072531557, 4.15059853721276827319737035449, 4.68350720223686741005719984071, 5.71215091283826514215439040376, 6.16824768783596798507203745713, 6.67980824846698397265124909736, 7.77073320880085002950943314440, 8.345479430020144932350275677635
