| L(s) = 1 | + 640. i·5-s − 907.·7-s + 8.11e3i·11-s + 4.79e4·13-s − 2.98e4i·17-s − 7.03e3·19-s − 3.51e5i·23-s − 1.93e4·25-s − 3.87e5i·29-s + 6.37e4·31-s − 5.81e5i·35-s + 2.45e6·37-s + 3.12e6i·41-s + 2.12e6·43-s − 2.22e6i·47-s + ⋯ |
| L(s) = 1 | + 1.02i·5-s − 0.377·7-s + 0.554i·11-s + 1.67·13-s − 0.356i·17-s − 0.0539·19-s − 1.25i·23-s − 0.0495·25-s − 0.548i·29-s + 0.0690·31-s − 0.387i·35-s + 1.31·37-s + 1.10i·41-s + 0.620·43-s − 0.455i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.281959479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.281959479\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 907.T \) |
| good | 5 | \( 1 - 640. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 8.11e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.79e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 2.98e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 7.03e3T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.51e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.87e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 6.37e4T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.45e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.12e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.12e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 2.22e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.76e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.12e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 4.08e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.48e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.73e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.58e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.14e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.80e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 2.69e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.47e8T + 7.83e15T^{2} \) |
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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
−10.77903859813936180559020962991, −9.940743016210434112740762086641, −8.852281717558473416778483590256, −7.76782548145579235483020649719, −6.63889775064842428024754756917, −6.07292631468561240177052566327, −4.47551123743654425029056684731, −3.35787516872234800686053050458, −2.36112868206598293785480295441, −0.863278887719923707765976609656,
0.67323425946412399544890278251, 1.53187903809915818717476687220, 3.24261889970357741864369212642, 4.20933147221606811793843092583, 5.52094202897250068023222731392, 6.26419553713769151131256990615, 7.69180605558201936016030234132, 8.711108316127010596454502578878, 9.229253361283294474393759599162, 10.55316329396242066671607037231
