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
−10.59523025583029202558354503309, −9.458419761601227879363089501348, −8.702602132672200820601415364440, −7.978727741737425187783778722772, −6.80379843620544208445214810698, −6.18573546364636573254799723821, −4.94150342410443692419104271924, −3.63324610500658274442201888781, −2.58247034454246733586331312893, −0.914534402024743180882876203600,
1.77118414331097806789375930088, 3.15071750733572713514677841518, 4.40520587979508663278659154208, 5.02499277950116262149259163091, 6.55876832554089240168348208687, 7.19367553515283518122311309949, 8.443935749541386460275208542013, 9.116124051667141690885045631184, 10.16106844341492498816097585141, 10.47943702209173908529826735682