L(s) = 1 | − 0.961·2-s − 1.98·3-s − 1.07·4-s + 3.39·5-s + 1.91·6-s + 7-s + 2.95·8-s + 0.949·9-s − 3.26·10-s + 4.59·11-s + 2.13·12-s − 0.961·14-s − 6.74·15-s − 0.692·16-s + 2.44·17-s − 0.913·18-s + 4.77·19-s − 3.64·20-s − 1.98·21-s − 4.41·22-s − 4.04·23-s − 5.87·24-s + 6.51·25-s + 4.07·27-s − 1.07·28-s − 3.20·29-s + 6.48·30-s + ⋯ |
L(s) = 1 | − 0.679·2-s − 1.14·3-s − 0.537·4-s + 1.51·5-s + 0.780·6-s + 0.377·7-s + 1.04·8-s + 0.316·9-s − 1.03·10-s + 1.38·11-s + 0.616·12-s − 0.256·14-s − 1.74·15-s − 0.173·16-s + 0.593·17-s − 0.215·18-s + 1.09·19-s − 0.815·20-s − 0.433·21-s − 0.941·22-s − 0.843·23-s − 1.19·24-s + 1.30·25-s + 0.784·27-s − 0.203·28-s − 0.595·29-s + 1.18·30-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
\(L(1)\) |
\(\approx\) |
\(0.9937802748\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9937802748\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.961T + 2T^{2} \) |
| 3 | \( 1 + 1.98T + 3T^{2} \) |
| 5 | \( 1 - 3.39T + 5T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 + 4.83T + 31T^{2} \) |
| 37 | \( 1 - 9.61T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 - 9.97T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 8.52T + 71T^{2} \) |
| 73 | \( 1 - 6.62T + 73T^{2} \) |
| 79 | \( 1 + 7.98T + 79T^{2} \) |
| 83 | \( 1 + 7.30T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 - 6.90T + 97T^{2} \) |
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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
−9.913263869164624752461396259624, −9.154368264840671151617353164424, −8.355627788104218330798810568753, −7.18491537744240699536183656859, −6.30029250256651544600159701995, −5.53126855686657912519561635346, −4.99316344253991083137328069065, −3.72683278404947439163704095345, −1.85259207077730388673580594500, −0.957939192848369187071185420165,
0.957939192848369187071185420165, 1.85259207077730388673580594500, 3.72683278404947439163704095345, 4.99316344253991083137328069065, 5.53126855686657912519561635346, 6.30029250256651544600159701995, 7.18491537744240699536183656859, 8.355627788104218330798810568753, 9.154368264840671151617353164424, 9.913263869164624752461396259624