| L(s) = 1 | − 5.43·2-s + 21.5·4-s + 21.1·5-s − 20.1·7-s − 73.7·8-s − 115.·10-s + 58.1·11-s − 34.3·13-s + 109.·14-s + 228.·16-s + 2.49·17-s − 34.4·19-s + 456.·20-s − 316.·22-s + 127.·23-s + 323.·25-s + 186.·26-s − 434.·28-s + 288.·29-s + 286.·31-s − 651.·32-s − 13.5·34-s − 427.·35-s + 135.·37-s + 187.·38-s − 1.56e3·40-s − 360.·41-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.69·4-s + 1.89·5-s − 1.08·7-s − 3.25·8-s − 3.64·10-s + 1.59·11-s − 0.733·13-s + 2.09·14-s + 3.56·16-s + 0.0355·17-s − 0.416·19-s + 5.10·20-s − 3.06·22-s + 1.15·23-s + 2.59·25-s + 1.41·26-s − 2.93·28-s + 1.84·29-s + 1.66·31-s − 3.59·32-s − 0.0683·34-s − 2.06·35-s + 0.603·37-s + 0.800·38-s − 6.17·40-s − 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.424081249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424081249\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 5.43T + 8T^{2} \) |
| 5 | \( 1 - 21.1T + 125T^{2} \) |
| 7 | \( 1 + 20.1T + 343T^{2} \) |
| 11 | \( 1 - 58.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.49T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 127.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 288.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 286.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 135.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 360.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 419.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 129.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 579.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 545.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 9.17T + 3.00e5T^{2} \) |
| 71 | \( 1 - 335.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 256.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 402.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 364.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 911.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 988.T + 9.12e5T^{2} \) |
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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
−8.851260733338508779471907906573, −8.423212036256066094929286626489, −6.90778875485304303860780779525, −6.56078452502326646649037314241, −6.28140690684535302343825874480, −5.01003799650473643103001750452, −3.14431278953356252425487503227, −2.47568828651999541602266199541, −1.50006856576716111614325422342, −0.77376287095516963841758785128,
0.77376287095516963841758785128, 1.50006856576716111614325422342, 2.47568828651999541602266199541, 3.14431278953356252425487503227, 5.01003799650473643103001750452, 6.28140690684535302343825874480, 6.56078452502326646649037314241, 6.90778875485304303860780779525, 8.423212036256066094929286626489, 8.851260733338508779471907906573
