L(s) = 1 | + (−1.41 − i)3-s + (−1.73 + 1.41i)5-s + (1.00 + 2.82i)9-s − 4.89·11-s + 4.89i·13-s + (3.86 − 0.267i)15-s − 3.46·17-s + 3.46i·19-s + 6i·23-s + (0.999 − 4.89i)25-s + (1.41 − 5.00i)27-s − 2.82i·29-s − 3.46i·31-s + (6.92 + 4.89i)33-s − 4.89i·37-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.577i)3-s + (−0.774 + 0.632i)5-s + (0.333 + 0.942i)9-s − 1.47·11-s + 1.35i·13-s + (0.997 − 0.0691i)15-s − 0.840·17-s + 0.794i·19-s + 1.25i·23-s + (0.199 − 0.979i)25-s + (0.272 − 0.962i)27-s − 0.525i·29-s − 0.622i·31-s + (1.20 + 0.852i)33-s − 0.805i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.169004 + 0.317631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.169004 + 0.317631i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 5 | \( 1 + (1.73 - 1.41i)T \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.29842972763067572273573034576, −11.42134135491692517718233535634, −10.89201704853482193869461871279, −9.796055159475602554161942082646, −8.206512529521974946875474537125, −7.41244561583080313598238892149, −6.54147319896468511900706437894, −5.33574215362576482465350029422, −4.04350852873106186643099601937, −2.23382302473330799143475228313,
0.30577859808705926005050869219, 3.14342414870859641184431479662, 4.70939093318512291498847350140, 5.23170425648756965449813761998, 6.67697981791199722123826300309, 7.958268692401170129599735329895, 8.806048196688570406636679769706, 10.19344748194443834927611484899, 10.78462466243120143156101306806, 11.72595960861378946355439975508