| L(s) = 1 | − 2·2-s + 4·4-s − 20·5-s − 7·7-s − 8·8-s + 40·10-s − 28·11-s + 3·13-s + 14·14-s + 16·16-s − 115·17-s − 106·19-s − 80·20-s + 56·22-s + 149·23-s + 275·25-s − 6·26-s − 28·28-s − 11·29-s + 81·31-s − 32·32-s + 230·34-s + 140·35-s + 2·37-s + 212·38-s + 160·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 0.353·8-s + 1.26·10-s − 0.767·11-s + 0.0640·13-s + 0.267·14-s + 1/4·16-s − 1.64·17-s − 1.27·19-s − 0.894·20-s + 0.542·22-s + 1.35·23-s + 11/5·25-s − 0.0452·26-s − 0.188·28-s − 0.0704·29-s + 0.469·31-s − 0.176·32-s + 1.16·34-s + 0.676·35-s + 0.00888·37-s + 0.905·38-s + 0.632·40-s + 0.0228·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4323902727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4323902727\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 3 T + p^{3} T^{2} \) |
| 17 | \( 1 + 115 T + p^{3} T^{2} \) |
| 19 | \( 1 + 106 T + p^{3} T^{2} \) |
| 23 | \( 1 - 149 T + p^{3} T^{2} \) |
| 29 | \( 1 + 11 T + p^{3} T^{2} \) |
| 31 | \( 1 - 81 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 139 T + p^{3} T^{2} \) |
| 47 | \( 1 + 474 T + p^{3} T^{2} \) |
| 53 | \( 1 - 531 T + p^{3} T^{2} \) |
| 59 | \( 1 - 817 T + p^{3} T^{2} \) |
| 61 | \( 1 - 498 T + p^{3} T^{2} \) |
| 67 | \( 1 - 793 T + p^{3} T^{2} \) |
| 71 | \( 1 + 853 T + p^{3} T^{2} \) |
| 73 | \( 1 - 490 T + p^{3} T^{2} \) |
| 79 | \( 1 + 330 T + p^{3} T^{2} \) |
| 83 | \( 1 - 404 T + p^{3} T^{2} \) |
| 89 | \( 1 + 831 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1424 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.09787333756271314713084431485, −10.13141431546169891997353992504, −8.700731738080499188462612161028, −8.390492531960356457422412841654, −7.23338818147748278309879480776, −6.61614109562118241796271810840, −4.86585636737649791720456208640, −3.80604172166824471399131796663, −2.54923711236853814952228633040, −0.45636643859399431664473716095,
0.45636643859399431664473716095, 2.54923711236853814952228633040, 3.80604172166824471399131796663, 4.86585636737649791720456208640, 6.61614109562118241796271810840, 7.23338818147748278309879480776, 8.390492531960356457422412841654, 8.700731738080499188462612161028, 10.13141431546169891997353992504, 11.09787333756271314713084431485
