L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s + 5-s + 1.61·6-s − 0.618·7-s − 2.23·8-s + 9-s + 1.61·10-s − 0.236·11-s + 0.618·12-s − 13-s − 1.00·14-s + 15-s − 4.85·16-s − 2.85·17-s + 1.61·18-s + 0.854·19-s + 0.618·20-s − 0.618·21-s − 0.381·22-s + 4.85·23-s − 2.23·24-s − 4·25-s − 1.61·26-s + 27-s − 0.381·28-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.447·5-s + 0.660·6-s − 0.233·7-s − 0.790·8-s + 0.333·9-s + 0.511·10-s − 0.0711·11-s + 0.178·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s − 1.21·16-s − 0.692·17-s + 0.381·18-s + 0.195·19-s + 0.138·20-s − 0.134·21-s − 0.0814·22-s + 1.01·23-s − 0.456·24-s − 0.800·25-s − 0.317·26-s + 0.192·27-s − 0.0721·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.140280936\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.140280936\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 5.61T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 5.94T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 - 2.61T + 73T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 - 9.85T + 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
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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
−12.96398085902552693312516278260, −12.10736179209522535271786520664, −10.85343415348044766005811362944, −9.540857999788061041666565340263, −8.816661258828038502276127464856, −7.30310767959864380296572794873, −6.10639120906328679343439521196, −4.97990707064660026254163588385, −3.77359646576267204480333738188, −2.49481129884255761010940407757,
2.49481129884255761010940407757, 3.77359646576267204480333738188, 4.97990707064660026254163588385, 6.10639120906328679343439521196, 7.30310767959864380296572794873, 8.816661258828038502276127464856, 9.540857999788061041666565340263, 10.85343415348044766005811362944, 12.10736179209522535271786520664, 12.96398085902552693312516278260