| L(s) = 1 | + 2.53·2-s − 1.59·4-s − 27.8·7-s − 24.2·8-s + 35.8·11-s + 27.3·13-s − 70.5·14-s − 48.6·16-s − 93.6·17-s + 135.·19-s + 90.6·22-s + 0.407·23-s + 69.2·26-s + 44.5·28-s + 194.·29-s − 96.7·31-s + 71.1·32-s − 236.·34-s + 186.·37-s + 343.·38-s − 53.6·41-s + 519.·43-s − 57.2·44-s + 1.03·46-s − 190.·47-s + 434.·49-s − 43.7·52-s + ⋯ |
| L(s) = 1 | + 0.894·2-s − 0.199·4-s − 1.50·7-s − 1.07·8-s + 0.981·11-s + 0.583·13-s − 1.34·14-s − 0.760·16-s − 1.33·17-s + 1.63·19-s + 0.878·22-s + 0.00369·23-s + 0.522·26-s + 0.300·28-s + 1.24·29-s − 0.560·31-s + 0.393·32-s − 1.19·34-s + 0.830·37-s + 1.46·38-s − 0.204·41-s + 1.84·43-s − 0.196·44-s + 0.00330·46-s − 0.591·47-s + 1.26·49-s − 0.116·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.142623978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142623978\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.53T + 8T^{2} \) |
| 7 | \( 1 + 27.8T + 343T^{2} \) |
| 11 | \( 1 - 35.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.407T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 96.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 186.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 53.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 519.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 190.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 533.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 472.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 327.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 78.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 707.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 344.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 98.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 448.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 472.T + 9.12e5T^{2} \) |
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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
−9.890422390154077645530412707886, −9.291629420785936884609927688183, −8.630251291050510103914448387956, −7.07946781292177395644379249829, −6.36417960133212813960406164758, −5.64283787608539005309343459761, −4.38137761457502180465217750064, −3.62093939766273638305106414824, −2.74439316099036633673140211219, −0.74066201650976485270336472157,
0.74066201650976485270336472157, 2.74439316099036633673140211219, 3.62093939766273638305106414824, 4.38137761457502180465217750064, 5.64283787608539005309343459761, 6.36417960133212813960406164758, 7.07946781292177395644379249829, 8.630251291050510103914448387956, 9.291629420785936884609927688183, 9.890422390154077645530412707886
