| L(s) = 1 | + 2.41i·2-s − 3.82·4-s − 2.82i·7-s − 4.41i·8-s + 2·11-s + i·13-s + 6.82·14-s + 2.99·16-s + 3.65i·17-s − 2.82·19-s + 4.82i·22-s − 4i·23-s − 2.41·26-s + 10.8i·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 1.70i·2-s − 1.91·4-s − 1.06i·7-s − 1.56i·8-s + 0.603·11-s + 0.277i·13-s + 1.82·14-s + 0.749·16-s + 0.886i·17-s − 0.648·19-s + 1.02i·22-s − 0.834i·23-s − 0.473·26-s + 2.04i·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5555333210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5555333210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 - 2.41iT - 2T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 3.65iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.65iT - 43T^{2} \) |
| 47 | \( 1 - 0.343iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 1.17iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 7.65iT - 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 + 7.65iT - 97T^{2} \) |
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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
−8.656782677360255251925750982150, −7.73960306963563367125796756252, −7.11240682526803810336635406528, −6.52880549318635270305676292691, −5.94627527729008650753817362560, −4.88990142572666480375565876894, −4.26090567206814788500256422340, −3.52418270641768189121223973661, −1.73433159174333204363285244415, −0.18145741592104828087080349016,
1.30646868951068850295115389795, 2.20765272160579122517219843451, 3.00344173497458103003440206696, 3.76241735436634338670244144707, 4.73327250281987917704841038767, 5.44944620604901684629876865597, 6.40162772608198095172884063951, 7.46521178338541765702949582608, 8.521435912112687195725768203621, 9.082997493864862003407060047763
