| L(s) = 1 | + 3·3-s − 7.81·5-s + 9·9-s + 42.7·11-s − 10.9·13-s − 23.4·15-s + 80.2·17-s + 112.·19-s − 70.1·23-s − 63.9·25-s + 27·27-s − 147.·29-s − 144.·31-s + 128.·33-s − 0.292·37-s − 32.8·39-s − 294.·41-s + 337.·43-s − 70.3·45-s + 104.·47-s + 240.·51-s − 143.·53-s − 334.·55-s + 338.·57-s + 520.·59-s + 399.·61-s + 85.5·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.698·5-s + 0.333·9-s + 1.17·11-s − 0.233·13-s − 0.403·15-s + 1.14·17-s + 1.36·19-s − 0.636·23-s − 0.511·25-s + 0.192·27-s − 0.941·29-s − 0.837·31-s + 0.676·33-s − 0.00130·37-s − 0.134·39-s − 1.12·41-s + 1.19·43-s − 0.232·45-s + 0.324·47-s + 0.661·51-s − 0.370·53-s − 0.819·55-s + 0.785·57-s + 1.14·59-s + 0.839·61-s + 0.163·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.538284234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.538284234\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 7.81T + 125T^{2} \) |
| 11 | \( 1 - 42.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 70.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 0.292T + 5.06e4T^{2} \) |
| 41 | \( 1 + 294.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 337.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 104.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 143.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 520.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 399.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 137.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 266.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 524.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 433.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 664.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 803.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 445.T + 9.12e5T^{2} \) |
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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
−9.460549700044920234105530461176, −8.559246540090970395303047508533, −7.64837811954482869581630387315, −7.24764178501476914137217167689, −6.04861014780150115173124488835, −5.07799864022512219590540198193, −3.78613268527502445886013362148, −3.49644518085256613536370642733, −2.00658154567375829179459031437, −0.821673458897834338794463154084,
0.821673458897834338794463154084, 2.00658154567375829179459031437, 3.49644518085256613536370642733, 3.78613268527502445886013362148, 5.07799864022512219590540198193, 6.04861014780150115173124488835, 7.24764178501476914137217167689, 7.64837811954482869581630387315, 8.559246540090970395303047508533, 9.460549700044920234105530461176
