L(s) = 1 | + 0.615·2-s + 9·3-s − 31.6·4-s + 61.9·5-s + 5.53·6-s − 209.·7-s − 39.1·8-s + 81·9-s + 38.1·10-s + 1.45·11-s − 284.·12-s + 351.·13-s − 128.·14-s + 557.·15-s + 987.·16-s + 689.·17-s + 49.8·18-s + 2.73e3·19-s − 1.95e3·20-s − 1.88e3·21-s + 0.895·22-s − 1.31e3·23-s − 352.·24-s + 711.·25-s + 216.·26-s + 729·27-s + 6.62e3·28-s + ⋯ |
L(s) = 1 | + 0.108·2-s + 0.577·3-s − 0.988·4-s + 1.10·5-s + 0.0628·6-s − 1.61·7-s − 0.216·8-s + 0.333·9-s + 0.120·10-s + 0.00362·11-s − 0.570·12-s + 0.576·13-s − 0.175·14-s + 0.639·15-s + 0.964·16-s + 0.578·17-s + 0.0362·18-s + 1.73·19-s − 1.09·20-s − 0.933·21-s + 0.000394·22-s − 0.516·23-s − 0.124·24-s + 0.227·25-s + 0.0627·26-s + 0.192·27-s + 1.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.161211866\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.161211866\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
good | 2 | \( 1 - 0.615T + 32T^{2} \) |
| 5 | \( 1 - 61.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 209.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 1.45T + 1.61e5T^{2} \) |
| 13 | \( 1 - 351.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 689.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 508.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.25e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.53e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.65e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 4.26e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.35e5T + 8.58e9T^{2} \) |
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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
−12.19652148076792358221733907555, −10.29546505742283704025098608503, −9.571013754296111425047454830042, −9.194166131776140881912040999251, −7.79012814704782905740070857240, −6.31471176896636702149528044085, −5.47265288674357907631909711682, −3.82506940322294436090286179966, −2.83310834183008786614299267827, −0.930176398529957473961550683274,
0.930176398529957473961550683274, 2.83310834183008786614299267827, 3.82506940322294436090286179966, 5.47265288674357907631909711682, 6.31471176896636702149528044085, 7.79012814704782905740070857240, 9.194166131776140881912040999251, 9.571013754296111425047454830042, 10.29546505742283704025098608503, 12.19652148076792358221733907555