| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 8.24·7-s + 2.82i·8-s + 3i·11-s − 13.4·13-s − 11.6i·14-s + 4.00·16-s + 16.2i·17-s + 24.9·19-s + 4.24·22-s − 23.4i·23-s + 19.0i·26-s − 16.4·28-s + 40.9i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.17·7-s + 0.353i·8-s + 0.272i·11-s − 1.03·13-s − 0.832i·14-s + 0.250·16-s + 0.955i·17-s + 1.31·19-s + 0.192·22-s − 1.02i·23-s + 0.733i·26-s − 0.588·28-s + 1.41i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.935247195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.935247195\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 8.24T + 49T^{2} \) |
| 11 | \( 1 - 3iT - 121T^{2} \) |
| 13 | \( 1 + 13.4T + 169T^{2} \) |
| 17 | \( 1 - 16.2iT - 289T^{2} \) |
| 19 | \( 1 - 24.9T + 361T^{2} \) |
| 23 | \( 1 + 23.4iT - 529T^{2} \) |
| 29 | \( 1 - 40.9iT - 841T^{2} \) |
| 31 | \( 1 + 13.2T + 961T^{2} \) |
| 37 | \( 1 - 14.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 14.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 23.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 17.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 95.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 81.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 89.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.08T + 5.32e3T^{2} \) |
| 79 | \( 1 + 1.81T + 6.24e3T^{2} \) |
| 83 | \( 1 + 109. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 40.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
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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
−9.463770297421126964260373674124, −8.731007799256755753039721331477, −7.83229377891365820473956668099, −7.23434676107452893561581595884, −5.89349128186290972754816713955, −4.94725183021342033320243209536, −4.38995550658986764929502188288, −3.14295192922114325334091519657, −2.08927727629000220512949354431, −1.09619806654172547432024920260,
0.64940926014753825891214978164, 2.09038390842733953905987536232, 3.40402289905226074543740111252, 4.64931860925284474355738326171, 5.18554952185008734668774302174, 6.00502648457968476996687544835, 7.38562061373541964168166456442, 7.50824598675740842446402545295, 8.442702687928955959444502677437, 9.445671927049195088519089602946
