L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)11-s − 2·13-s + (−2 + 3.46i)19-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s − 0.999·27-s + 6·29-s + (4 + 6.92i)31-s + (−3 + 5.19i)33-s + (−1 + 1.73i)37-s + (−1 − 1.73i)39-s − 12·41-s − 4·43-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.166 + 0.288i)9-s + (0.904 + 1.56i)11-s − 0.554·13-s + (−0.458 + 0.794i)19-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s − 0.192·27-s + 1.11·29-s + (0.718 + 1.24i)31-s + (−0.522 + 0.904i)33-s + (−0.164 + 0.284i)37-s + (−0.160 − 0.277i)39-s − 1.87·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24517 + 0.947274i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24517 + 0.947274i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.47943702209173908529826735682, −10.16106844341492498816097585141, −9.116124051667141690885045631184, −8.443935749541386460275208542013, −7.19367553515283518122311309949, −6.55876832554089240168348208687, −5.02499277950116262149259163091, −4.40520587979508663278659154208, −3.15071750733572713514677841518, −1.77118414331097806789375930088,
0.914534402024743180882876203600, 2.58247034454246733586331312893, 3.63324610500658274442201888781, 4.94150342410443692419104271924, 6.18573546364636573254799723821, 6.80379843620544208445214810698, 7.978727741737425187783778722772, 8.702602132672200820601415364440, 9.458419761601227879363089501348, 10.59523025583029202558354503309