L(s) = 1 | − 0.414·2-s − 1.82·4-s + 5-s − 7-s + 1.58·8-s − 0.414·10-s − 2.41·11-s − 0.414·13-s + 0.414·14-s + 3·16-s − 0.828·17-s + 4.41·19-s − 1.82·20-s + 0.999·22-s + 4.82·23-s + 25-s + 0.171·26-s + 1.82·28-s − 4·29-s + 6·31-s − 4.41·32-s + 0.343·34-s − 35-s + 8.48·37-s − 1.82·38-s + 1.58·40-s − 2.17·41-s + ⋯ |
L(s) = 1 | − 0.292·2-s − 0.914·4-s + 0.447·5-s − 0.377·7-s + 0.560·8-s − 0.130·10-s − 0.727·11-s − 0.114·13-s + 0.110·14-s + 0.750·16-s − 0.200·17-s + 1.01·19-s − 0.408·20-s + 0.213·22-s + 1.00·23-s + 0.200·25-s + 0.0336·26-s + 0.345·28-s − 0.742·29-s + 1.07·31-s − 0.780·32-s + 0.0588·34-s − 0.169·35-s + 1.39·37-s − 0.296·38-s + 0.250·40-s − 0.339·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.069464015\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.069464015\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 0.414T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 4.65T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 2.17T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−9.791268141694709254995948243740, −9.418679887553456665556912496201, −8.440810768690363647788760765134, −7.68924573585406078381363536238, −6.69514942262324026107291338691, −5.52166826004903773727527928290, −4.91871515518784025867642200364, −3.72785437125309116310886621011, −2.56236852206653238634458799461, −0.873412705238580479693108341594,
0.873412705238580479693108341594, 2.56236852206653238634458799461, 3.72785437125309116310886621011, 4.91871515518784025867642200364, 5.52166826004903773727527928290, 6.69514942262324026107291338691, 7.68924573585406078381363536238, 8.440810768690363647788760765134, 9.418679887553456665556912496201, 9.791268141694709254995948243740