| L(s) = 1 | − 9.82·5-s + 14.1·11-s − 26.1·13-s + 78.5·17-s + 73.1·19-s + 96·23-s − 28.4·25-s − 173.·29-s + 67.2·31-s − 301.·37-s − 472.·41-s − 463.·43-s + 91.1·47-s + 163.·53-s − 139.·55-s + 600.·59-s + 571.·61-s + 256.·65-s − 539.·67-s + 1.06e3·71-s − 442.·73-s − 45.7·79-s + 686.·83-s − 772.·85-s − 660.·89-s − 718.·95-s + 658.·97-s + ⋯ |
| L(s) = 1 | − 0.878·5-s + 0.388·11-s − 0.557·13-s + 1.12·17-s + 0.883·19-s + 0.870·23-s − 0.227·25-s − 1.10·29-s + 0.389·31-s − 1.34·37-s − 1.79·41-s − 1.64·43-s + 0.283·47-s + 0.423·53-s − 0.341·55-s + 1.32·59-s + 1.19·61-s + 0.489·65-s − 0.983·67-s + 1.77·71-s − 0.709·73-s − 0.0650·79-s + 0.907·83-s − 0.985·85-s − 0.786·89-s − 0.776·95-s + 0.689·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.557025076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557025076\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 9.82T + 125T^{2} \) |
| 11 | \( 1 - 14.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96T + 1.21e4T^{2} \) |
| 29 | \( 1 + 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 472.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 91.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 600.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 571.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 539.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 45.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 686.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 660.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 658.T + 9.12e5T^{2} \) |
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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
−8.843936182139608526676785779832, −8.112555006259609676034533568755, −7.33168268264071741847018725930, −6.80119878451450598183650776440, −5.51373486549161232979806143276, −4.93410294374654109307218445959, −3.71364937799688449384097425455, −3.23991109019821957333915354255, −1.79368285463605973708956418384, −0.59386709720937646950305169960,
0.59386709720937646950305169960, 1.79368285463605973708956418384, 3.23991109019821957333915354255, 3.71364937799688449384097425455, 4.93410294374654109307218445959, 5.51373486549161232979806143276, 6.80119878451450598183650776440, 7.33168268264071741847018725930, 8.112555006259609676034533568755, 8.843936182139608526676785779832
