| L(s) = 1 | + 0.983·2-s − 1.57·3-s − 1.03·4-s + 0.398·5-s − 1.54·6-s + 7-s − 2.98·8-s − 0.529·9-s + 0.391·10-s + 4.24·11-s + 1.62·12-s + 0.983·14-s − 0.626·15-s − 0.870·16-s − 5.10·17-s − 0.521·18-s + 2.12·19-s − 0.411·20-s − 1.57·21-s + 4.17·22-s + 2.19·23-s + 4.68·24-s − 4.84·25-s + 5.54·27-s − 1.03·28-s + 2.90·29-s − 0.616·30-s + ⋯ |
| L(s) = 1 | + 0.695·2-s − 0.907·3-s − 0.516·4-s + 0.178·5-s − 0.631·6-s + 0.377·7-s − 1.05·8-s − 0.176·9-s + 0.123·10-s + 1.27·11-s + 0.468·12-s + 0.262·14-s − 0.161·15-s − 0.217·16-s − 1.23·17-s − 0.122·18-s + 0.487·19-s − 0.0919·20-s − 0.342·21-s + 0.889·22-s + 0.457·23-s + 0.957·24-s − 0.968·25-s + 1.06·27-s − 0.195·28-s + 0.540·29-s − 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.404314680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404314680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.983T + 2T^{2} \) |
| 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 - 0.398T + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 - 2.12T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 - 2.20T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 5.07T + 41T^{2} \) |
| 43 | \( 1 + 0.328T + 43T^{2} \) |
| 47 | \( 1 - 6.62T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 + 1.75T + 71T^{2} \) |
| 73 | \( 1 + 6.11T + 73T^{2} \) |
| 79 | \( 1 - 4.20T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
| 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
−9.670439186913023646321710285251, −9.045044314249960743849657616193, −8.251044393459603589984741469891, −6.92651110490940308332017169996, −6.14176567259388721191823150381, −5.55565809833974817796280170965, −4.56960104983876190232929051452, −3.99814063522320007956533337897, −2.60813802908636498645321309946, −0.856550874745753894241813130406,
0.856550874745753894241813130406, 2.60813802908636498645321309946, 3.99814063522320007956533337897, 4.56960104983876190232929051452, 5.55565809833974817796280170965, 6.14176567259388721191823150381, 6.92651110490940308332017169996, 8.251044393459603589984741469891, 9.045044314249960743849657616193, 9.670439186913023646321710285251
