L(s) = 1 | + 1.82i·2-s + 4.68·4-s − 15.0·5-s + 23.0i·8-s − 27.4i·10-s − 9.89i·11-s − 67.8i·13-s − 4.54·16-s + 70.1·17-s − 61.4i·19-s − 70.7·20-s + 18.0·22-s + 131. i·23-s + 102.·25-s + 123.·26-s + ⋯ |
L(s) = 1 | + 0.643i·2-s + 0.585·4-s − 1.34·5-s + 1.02i·8-s − 0.868i·10-s − 0.271i·11-s − 1.44i·13-s − 0.0710·16-s + 1.00·17-s − 0.742i·19-s − 0.790·20-s + 0.174·22-s + 1.19i·23-s + 0.821·25-s + 0.932·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.739260588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.739260588\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.82iT - 8T^{2} \) |
| 5 | \( 1 + 15.0T + 125T^{2} \) |
| 11 | \( 1 + 9.89iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 67.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 70.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 131. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 76.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 348.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 539.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 530. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 542.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 134. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 320.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 416. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 545. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 322.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 885.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 739. iT - 9.12e5T^{2} \) |
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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
−10.98411120346213101374696047814, −9.890999103652848538526418334611, −8.488926408543118205116238303403, −7.74680516935687237518336428667, −7.36199610333523379548817174242, −6.03859816855554030566109032946, −5.19306180218415961152716827817, −3.75541897116314389120448540590, −2.75226741232278185299899707067, −0.71849247831558934727287399603,
1.01149850023005358664082557902, 2.45262471232157360928187675231, 3.71570286506940199834339276914, 4.39859412438692122421031825411, 6.08454570215618172082961422929, 7.14359388496003971060654543031, 7.76408187373839826792904976656, 8.926821899379578071867905227792, 9.989364061384210179964632427537, 10.88786861473611051611520798819