L(s) = 1 | + (−0.340 − 1.26i)2-s + (−0.224 + 1.71i)3-s + (0.236 − 0.136i)4-s + (2.25 − 0.299i)6-s + (−2.32 − 1.25i)7-s + (−2.11 − 2.11i)8-s + (−2.89 − 0.769i)9-s + (−3.38 + 1.95i)11-s + (0.181 + 0.436i)12-s + (1.56 − 1.56i)13-s + (−0.807 + 3.38i)14-s + (−1.69 + 2.92i)16-s + (−2.58 − 0.693i)17-s + (0.00887 + 3.94i)18-s + (−1.61 − 0.930i)19-s + ⋯ |
L(s) = 1 | + (−0.240 − 0.897i)2-s + (−0.129 + 0.991i)3-s + (0.118 − 0.0681i)4-s + (0.921 − 0.122i)6-s + (−0.879 − 0.476i)7-s + (−0.746 − 0.746i)8-s + (−0.966 − 0.256i)9-s + (−1.01 + 0.588i)11-s + (0.0523 + 0.125i)12-s + (0.434 − 0.434i)13-s + (−0.215 + 0.903i)14-s + (−0.422 + 0.731i)16-s + (−0.627 − 0.168i)17-s + (0.00209 + 0.929i)18-s + (−0.369 − 0.213i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0113026 + 0.327953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0113026 + 0.327953i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.224 - 1.71i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.32 + 1.25i)T \) |
good | 2 | \( 1 + (0.340 + 1.26i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (3.38 - 1.95i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.56 + 1.56i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.58 + 0.693i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.61 + 0.930i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.38 - 0.638i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 0.513T + 29T^{2} \) |
| 31 | \( 1 + (4.29 + 7.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.60 + 1.77i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 0.308iT - 41T^{2} \) |
| 43 | \( 1 + (7.60 - 7.60i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.36 + 5.10i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.498 + 1.85i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.259 + 0.448i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.55 - 4.42i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.34 - 8.74i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 + (2.79 + 0.749i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.37 - 2.52i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.16 - 9.16i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.67 + 9.82i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.81 - 6.81i)T + 97iT^{2} \) |
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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
−10.43051192545286883691323135210, −9.784872965990101663477759950006, −9.150588809289465920734808619437, −7.85931646202411564618620218528, −6.57972331652556699580350957907, −5.70047920966088549471760909317, −4.37036790288760557479838029532, −3.37730493662064770975594357463, −2.38379043650658358146384070178, −0.18811273569908813014108717593,
2.20067169253495732133525350917, 3.25517658577993861269543304393, 5.28933278382090763312648276315, 6.16824915568954099228068766804, 6.65616804053916131243573624700, 7.65320811195995684440132837276, 8.449992179071592831725296882513, 9.065666677693792294496066948418, 10.53395887328987732132280034885, 11.40293901031081724623804219263