| L(s) = 1 | − 1.95·2-s − 0.296·3-s + 1.82·4-s − 5-s + 0.580·6-s − 3.56·7-s + 0.340·8-s − 2.91·9-s + 1.95·10-s + 5.56·11-s − 0.541·12-s − 5.26·13-s + 6.96·14-s + 0.296·15-s − 4.31·16-s + 1.40·17-s + 5.69·18-s − 1.82·20-s + 1.05·21-s − 10.8·22-s − 6.96·23-s − 0.101·24-s + 25-s + 10.3·26-s + 1.75·27-s − 6.50·28-s − 1.40·29-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 0.171·3-s + 0.912·4-s − 0.447·5-s + 0.237·6-s − 1.34·7-s + 0.120·8-s − 0.970·9-s + 0.618·10-s + 1.67·11-s − 0.156·12-s − 1.46·13-s + 1.86·14-s + 0.0766·15-s − 1.07·16-s + 0.341·17-s + 1.34·18-s − 0.408·20-s + 0.230·21-s − 2.31·22-s − 1.45·23-s − 0.0206·24-s + 0.200·25-s + 2.02·26-s + 0.337·27-s − 1.22·28-s − 0.261·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2706203068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2706203068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 3 | \( 1 + 0.296T + 3T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 23 | \( 1 + 6.96T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 + 8.26T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 6.59T + 73T^{2} \) |
| 79 | \( 1 + 5.47T + 79T^{2} \) |
| 83 | \( 1 + 4.15T + 83T^{2} \) |
| 89 | \( 1 - 9.23T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−9.503933080320324531375302096166, −8.574408233450999003676990007059, −7.909510686053589640759086683593, −6.87578184374898040401761781420, −6.55467413009441385029304705225, −5.39115809748685932343941940298, −4.11937921977363612826867595336, −3.21593653361583710890314896059, −1.97221363531484072848400416088, −0.42114664627238455157593270570,
0.42114664627238455157593270570, 1.97221363531484072848400416088, 3.21593653361583710890314896059, 4.11937921977363612826867595336, 5.39115809748685932343941940298, 6.55467413009441385029304705225, 6.87578184374898040401761781420, 7.909510686053589640759086683593, 8.574408233450999003676990007059, 9.503933080320324531375302096166
