L(s) = 1 | − 2.24i·2-s + (−1.79 − 1.79i)3-s − 3.04·4-s − 1.17i·5-s + (−4.03 + 4.03i)6-s + (−0.707 − 0.707i)7-s + 2.34i·8-s + 3.43i·9-s − 2.62·10-s + (−2.44 − 2.44i)11-s + (5.46 + 5.46i)12-s + (4.67 + 4.67i)13-s + (−1.58 + 1.58i)14-s + (−2.09 + 2.09i)15-s − 0.813·16-s + (0.749 − 0.749i)17-s + ⋯ |
L(s) = 1 | − 1.58i·2-s + (−1.03 − 1.03i)3-s − 1.52·4-s − 0.523i·5-s + (−1.64 + 1.64i)6-s + (−0.267 − 0.267i)7-s + 0.830i·8-s + 1.14i·9-s − 0.831·10-s + (−0.737 − 0.737i)11-s + (1.57 + 1.57i)12-s + (1.29 + 1.29i)13-s + (−0.424 + 0.424i)14-s + (−0.542 + 0.542i)15-s − 0.203·16-s + (0.181 − 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0482 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.441430 + 0.463251i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.441430 + 0.463251i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.707 + 0.707i)T \) |
| 41 | \( 1 + (3.04 + 5.63i)T \) |
good | 2 | \( 1 + 2.24iT - 2T^{2} \) |
| 3 | \( 1 + (1.79 + 1.79i)T + 3iT^{2} \) |
| 5 | \( 1 + 1.17iT - 5T^{2} \) |
| 11 | \( 1 + (2.44 + 2.44i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.67 - 4.67i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.749 + 0.749i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.52 + 4.52i)T - 19iT^{2} \) |
| 23 | \( 1 + 5.77T + 23T^{2} \) |
| 29 | \( 1 + (1.42 + 1.42i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 43 | \( 1 + 1.38iT - 43T^{2} \) |
| 47 | \( 1 + (5.85 - 5.85i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.04 + 7.04i)T + 53iT^{2} \) |
| 59 | \( 1 - 6.17T + 59T^{2} \) |
| 61 | \( 1 - 1.38iT - 61T^{2} \) |
| 67 | \( 1 + (3.26 - 3.26i)T - 67iT^{2} \) |
| 71 | \( 1 + (5.01 + 5.01i)T + 71iT^{2} \) |
| 73 | \( 1 + 6.15iT - 73T^{2} \) |
| 79 | \( 1 + (5.75 + 5.75i)T + 79iT^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + (0.0484 + 0.0484i)T + 89iT^{2} \) |
| 97 | \( 1 + (9.56 - 9.56i)T - 97iT^{2} \) |
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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
−11.37111070589092231956742046661, −10.65143781672749555744911122441, −9.467637510087555622468119559688, −8.492186039477856624496663264543, −7.06579737709572914008744921300, −6.01666916586281864384188154891, −4.78934685802941744600043202801, −3.39150529237696436794961543035, −1.72946901726695337367994250956, −0.56428579520509450016741112550,
3.52030149064085968435654165616, 4.92282712678538168661941311056, 5.69001564672757136828763938954, 6.29701254806719344379511312509, 7.58269578622135954417822625999, 8.422187771031606633828977198624, 9.844600243329206838151760423504, 10.37252182584577348586386082592, 11.38817095184673359486415203117, 12.58103168518219057176432379670