| L(s) = 1 | − 2.41·2-s + 0.537·3-s + 3.85·4-s − 1.29·6-s − 3.18·7-s − 4.49·8-s − 2.71·9-s + 4.15·11-s + 2.07·12-s + 2.07·13-s + 7.71·14-s + 3.15·16-s + 5.79·17-s + 6.56·18-s − 19-s − 1.71·21-s − 10.0·22-s − 2.60·23-s − 2.41·24-s − 5.01·26-s − 3.06·27-s − 12.2·28-s + 6·29-s + 2.59·31-s + 1.34·32-s + 2.23·33-s − 14.0·34-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.310·3-s + 1.92·4-s − 0.530·6-s − 1.20·7-s − 1.58·8-s − 0.903·9-s + 1.25·11-s + 0.597·12-s + 0.574·13-s + 2.06·14-s + 0.788·16-s + 1.40·17-s + 1.54·18-s − 0.229·19-s − 0.373·21-s − 2.14·22-s − 0.543·23-s − 0.492·24-s − 0.982·26-s − 0.590·27-s − 2.32·28-s + 1.11·29-s + 0.466·31-s + 0.237·32-s + 0.388·33-s − 2.40·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6233164250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6233164250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 0.537T + 3T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 2.59T + 31T^{2} \) |
| 37 | \( 1 - 4.30T + 37T^{2} \) |
| 41 | \( 1 + 0.599T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 + 8.75T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 2.72T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 2.07T + 97T^{2} \) |
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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
−10.64445523055575107706477762206, −9.870096958604981146649104114225, −9.189812703247826705310420328254, −8.550562170645793191322485778855, −7.66123923904716129602834200565, −6.56385841313150541200304637111, −5.94699259565334212563688337217, −3.74002195520786417313797068315, −2.62838815475436065308179961045, −0.937803168317371011469960140801,
0.937803168317371011469960140801, 2.62838815475436065308179961045, 3.74002195520786417313797068315, 5.94699259565334212563688337217, 6.56385841313150541200304637111, 7.66123923904716129602834200565, 8.550562170645793191322485778855, 9.189812703247826705310420328254, 9.870096958604981146649104114225, 10.64445523055575107706477762206
