| L(s) = 1 | − 0.430·2-s + 3.17·3-s − 1.81·4-s − 5-s − 1.36·6-s + 2.65·7-s + 1.64·8-s + 7.05·9-s + 0.430·10-s + 3.22·11-s − 5.75·12-s − 4.47·13-s − 1.14·14-s − 3.17·15-s + 2.92·16-s − 4.30·17-s − 3.03·18-s + 7.19·19-s + 1.81·20-s + 8.40·21-s − 1.38·22-s − 0.762·23-s + 5.20·24-s + 25-s + 1.92·26-s + 12.8·27-s − 4.81·28-s + ⋯ |
| L(s) = 1 | − 0.304·2-s + 1.83·3-s − 0.907·4-s − 0.447·5-s − 0.557·6-s + 1.00·7-s + 0.580·8-s + 2.35·9-s + 0.136·10-s + 0.972·11-s − 1.66·12-s − 1.24·13-s − 0.304·14-s − 0.818·15-s + 0.730·16-s − 1.04·17-s − 0.715·18-s + 1.64·19-s + 0.405·20-s + 1.83·21-s − 0.295·22-s − 0.159·23-s + 1.06·24-s + 0.200·25-s + 0.377·26-s + 2.47·27-s − 0.909·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.377476682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.377476682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.430T + 2T^{2} \) |
| 3 | \( 1 - 3.17T + 3T^{2} \) |
| 7 | \( 1 - 2.65T + 7T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + 0.762T + 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 - 9.63T + 31T^{2} \) |
| 37 | \( 1 - 0.950T + 37T^{2} \) |
| 41 | \( 1 - 9.26T + 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 4.09T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 1.32T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 4.67T + 83T^{2} \) |
| 89 | \( 1 - 4.03T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−9.510763091418324063395156209684, −8.927374094131516286277503244651, −8.138142258601865704040675080183, −7.70924101859232560804877543283, −6.93194917749538933697107377944, −5.06553653331233064952186738533, −4.36249009466159222119795834248, −3.63407162158171103174271146162, −2.44830066074141904339152071917, −1.26984330013292310263765969034,
1.26984330013292310263765969034, 2.44830066074141904339152071917, 3.63407162158171103174271146162, 4.36249009466159222119795834248, 5.06553653331233064952186738533, 6.93194917749538933697107377944, 7.70924101859232560804877543283, 8.138142258601865704040675080183, 8.927374094131516286277503244651, 9.510763091418324063395156209684
