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
\(L(1)\) |
\(\approx\) |
\(1.361848853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.361848853\) |
\(L(\frac{3}{2})\) |
|
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
−8.137642631571338413830685365365, −7.83441116956573254896172881395, −6.84388822764113622567687075784, −6.10632496998768174929538469577, −5.81327067346947768112851415532, −4.64496699551388641936262505747, −3.67013589589753772891413350354, −3.37152793510649895902444211951, −2.17137013844230969225548366816, −1.29021626087301612774485678751,
0.36430875051775939586685352683, 1.31094906334666250221423650975, 2.58550167368898605666903843000, 3.12896348996832925208357201810, 4.49344539033291363606646753540, 4.74289110609783343389073807103, 5.72699839924404140538497171094, 6.18966099046153864401364786175, 7.22203527596218426857924025139, 7.909628034555338269874771757407