L(s) = 1 | + (−0.0847 − 0.0489i)2-s + (−5.17 − 0.490i)3-s + (−3.99 − 6.91i)4-s + (9.06 + 15.6i)5-s + (0.414 + 0.294i)6-s + (12.7 + 13.4i)7-s + 1.56i·8-s + (26.5 + 5.07i)9-s − 1.77i·10-s + (32.0 + 18.5i)11-s + (17.2 + 37.7i)12-s + (−16.3 + 9.44i)13-s + (−0.423 − 1.76i)14-s + (−39.1 − 85.6i)15-s + (−31.8 + 55.2i)16-s − 62.5·17-s + ⋯ |
L(s) = 1 | + (−0.0299 − 0.0173i)2-s + (−0.995 − 0.0944i)3-s + (−0.499 − 0.864i)4-s + (0.810 + 1.40i)5-s + (0.0282 + 0.0200i)6-s + (0.688 + 0.725i)7-s + 0.0691i·8-s + (0.982 + 0.188i)9-s − 0.0561i·10-s + (0.878 + 0.507i)11-s + (0.415 + 0.908i)12-s + (−0.349 + 0.201i)13-s + (−0.00808 − 0.0336i)14-s + (−0.674 − 1.47i)15-s + (−0.498 + 0.862i)16-s − 0.891·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.971533 + 0.480956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.971533 + 0.480956i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (5.17 + 0.490i)T \) |
| 7 | \( 1 + (-12.7 - 13.4i)T \) |
good | 2 | \( 1 + (0.0847 + 0.0489i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-9.06 - 15.6i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-32.0 - 18.5i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (16.3 - 9.44i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 62.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-140. + 81.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-82.5 - 47.6i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-110. + 63.8i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 378.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-99.1 - 171. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-160. + 278. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-79.3 + 137. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 191. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (106. + 185. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (190. + 110. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-68.2 - 118. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 458. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 967. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-298. + 516. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-180. + 313. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 35.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.15e3 + 664. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.65824456885929173969741973893, −13.76562950215526166385856017888, −12.20370316910533525145840920070, −11.00346890108549914437757426968, −10.29610110051044832740045139926, −9.137743715627905258186427074496, −6.88010687974091640042812453652, −6.04700371295315449287247004729, −4.76797096637248047879095115633, −1.87832513603631104453891119308,
0.962658406447336003043529641479, 4.34255088887471091781662882816, 5.19175296937739259225732524127, 6.96559325325041144428388622619, 8.588292615387994117241072876879, 9.500082782817463994918746810033, 11.08238935376811965653907770348, 12.16217578031367397755189532477, 13.13012383771828051200600650491, 13.85448589421654221760514879611