L(s) = 1 | + 0.332i·2-s − 1.45·3-s + 1.88·4-s − 1.44i·5-s − 0.485i·6-s + 1.29i·8-s − 0.868·9-s + 0.480·10-s + 5.95i·11-s − 2.75·12-s + (−1.88 + 3.07i)13-s + 2.11i·15-s + 3.34·16-s + 4.32·17-s − 0.288i·18-s − 1.95i·19-s + ⋯ |
L(s) = 1 | + 0.235i·2-s − 0.842·3-s + 0.944·4-s − 0.646i·5-s − 0.198i·6-s + 0.457i·8-s − 0.289·9-s + 0.151·10-s + 1.79i·11-s − 0.796·12-s + (−0.524 + 0.851i)13-s + 0.544i·15-s + 0.837·16-s + 1.04·17-s − 0.0680i·18-s − 0.449i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13591 + 0.634778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13591 + 0.634778i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (1.88 - 3.07i)T \) |
good | 2 | \( 1 - 0.332iT - 2T^{2} \) |
| 3 | \( 1 + 1.45T + 3T^{2} \) |
| 5 | \( 1 + 1.44iT - 5T^{2} \) |
| 11 | \( 1 - 5.95iT - 11T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + 1.95iT - 19T^{2} \) |
| 23 | \( 1 - 0.540T + 23T^{2} \) |
| 29 | \( 1 - 7.15T + 29T^{2} \) |
| 31 | \( 1 - 6.10iT - 31T^{2} \) |
| 37 | \( 1 + 8.02iT - 37T^{2} \) |
| 41 | \( 1 - 7.55iT - 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 - 6.26iT - 47T^{2} \) |
| 53 | \( 1 + 2.77T + 53T^{2} \) |
| 59 | \( 1 + 0.851iT - 59T^{2} \) |
| 61 | \( 1 + 6.77T + 61T^{2} \) |
| 67 | \( 1 + 0.987iT - 67T^{2} \) |
| 71 | \( 1 - 3.76iT - 71T^{2} \) |
| 73 | \( 1 + 9.13iT - 73T^{2} \) |
| 79 | \( 1 + 0.131T + 79T^{2} \) |
| 83 | \( 1 + 2.66iT - 83T^{2} \) |
| 89 | \( 1 - 9.71iT - 89T^{2} \) |
| 97 | \( 1 + 6.58iT - 97T^{2} \) |
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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
−10.76759749387299831148042351354, −9.995667697773121189163208891727, −9.039591501032668971171296239828, −7.86911492181366299840391705603, −7.00687557817289414718758978119, −6.36135850891847599421596454538, −5.17028809596356484969271436820, −4.64189046275151316793589964348, −2.80176376661513841558521725495, −1.46677842796416818095654918085,
0.836801234523705902849339999878, 2.79691548136839716130964532491, 3.37218959300406712897258314511, 5.27604471252017865538041657587, 5.98351205188568006191595046481, 6.62109160248313810079433703780, 7.74698328969699066380130863980, 8.533044987116755792264629488565, 10.14548881952730059632860822464, 10.51126710954249488036774628335