L(s) = 1 | + (0.623 − 0.218i)2-s + (−0.433 − 0.900i)3-s + (−0.440 + 0.351i)4-s + (−0.467 − 0.467i)6-s + (−0.549 + 0.874i)8-s + (−0.623 + 0.781i)9-s + (0.507 + 0.244i)12-s + (1.75 + 0.400i)13-s + (−0.0263 + 0.115i)16-s + (−0.218 + 0.623i)18-s + (−0.433 − 0.900i)23-s + (1.02 + 0.115i)24-s + (0.623 + 0.781i)25-s + (1.18 − 0.133i)26-s + (0.974 + 0.222i)27-s + ⋯ |
L(s) = 1 | + (0.623 − 0.218i)2-s + (−0.433 − 0.900i)3-s + (−0.440 + 0.351i)4-s + (−0.467 − 0.467i)6-s + (−0.549 + 0.874i)8-s + (−0.623 + 0.781i)9-s + (0.507 + 0.244i)12-s + (1.75 + 0.400i)13-s + (−0.0263 + 0.115i)16-s + (−0.218 + 0.623i)18-s + (−0.433 − 0.900i)23-s + (1.02 + 0.115i)24-s + (0.623 + 0.781i)25-s + (1.18 − 0.133i)26-s + (0.974 + 0.222i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.226739208\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.226739208\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.433 + 0.900i)T \) |
| 23 | \( 1 + (0.433 + 0.900i)T \) |
| 29 | \( 1 + (-0.781 + 0.623i)T \) |
good | 2 | \( 1 + (-0.623 + 0.218i)T + (0.781 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 13 | \( 1 + (-1.75 - 0.400i)T + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 31 | \( 1 + (-0.467 - 1.33i)T + (-0.781 + 0.623i)T^{2} \) |
| 37 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 41 | \( 1 + (-1.19 - 1.19i)T + iT^{2} \) |
| 43 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 47 | \( 1 + (-0.189 + 0.119i)T + (0.433 - 0.900i)T^{2} \) |
| 53 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 + 1.24iT - T^{2} \) |
| 61 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 67 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 71 | \( 1 + (0.193 - 0.846i)T + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.0739 - 0.211i)T + (-0.781 - 0.623i)T^{2} \) |
| 79 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 83 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 97 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
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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
−9.000835305856390752837972334329, −8.464282101476009313973701699806, −7.87434394741036059009435495464, −6.69243689351507808966052872906, −6.19517488205074798333029889048, −5.27874241710077933171054471342, −4.46015617265511729932911200862, −3.48783870329856867173435881062, −2.53677183602777746486504616259, −1.21559059870066960333625837851,
0.990033349502577949822383483533, 3.00070842589473426903744912856, 3.91748638090881408048105335966, 4.41119120010882821349459674402, 5.45963438312379762172017224710, 5.96235092735779170368530048084, 6.57345229586693691346088768678, 7.927996332153613938930264579194, 8.861998914212730967873597602181, 9.297906404115256869179441245634