| L(s) = 1 | − 5·5-s − 10.2·7-s − 44.3·11-s − 50.8·13-s − 25.2·17-s + 31.3·19-s + 76.1·23-s + 25·25-s − 156.·29-s − 134.·31-s + 51.2·35-s + 81.2·37-s − 326.·41-s − 422.·43-s − 452.·47-s − 237.·49-s − 98.1·53-s + 221.·55-s + 540.·59-s + 522.·61-s + 254.·65-s + 129.·67-s + 26.6·71-s − 147.·73-s + 454.·77-s + 1.08e3·79-s + 594.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.553·7-s − 1.21·11-s − 1.08·13-s − 0.360·17-s + 0.378·19-s + 0.690·23-s + 0.200·25-s − 0.998·29-s − 0.779·31-s + 0.247·35-s + 0.360·37-s − 1.24·41-s − 1.49·43-s − 1.40·47-s − 0.693·49-s − 0.254·53-s + 0.543·55-s + 1.19·59-s + 1.09·61-s + 0.485·65-s + 0.235·67-s + 0.0444·71-s − 0.237·73-s + 0.673·77-s + 1.55·79-s + 0.786·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6085115216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6085115216\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 10.2T + 343T^{2} \) |
| 11 | \( 1 + 44.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 25.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 76.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 134.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 81.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 326.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 452.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 98.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 540.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 522.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 129.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 26.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 147.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 594.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 592.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 666.T + 9.12e5T^{2} \) |
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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
−8.640069145007169692386781121596, −7.898542612646149894491172648043, −7.20062389627834473503517601623, −6.54525637227768425840281942153, −5.28573482638398320642356962815, −4.94004841203514836853986461587, −3.66395363599345291434115072331, −2.92747259777681459064573381528, −1.93071564174774239903872652348, −0.33296650176582609211933709773,
0.33296650176582609211933709773, 1.93071564174774239903872652348, 2.92747259777681459064573381528, 3.66395363599345291434115072331, 4.94004841203514836853986461587, 5.28573482638398320642356962815, 6.54525637227768425840281942153, 7.20062389627834473503517601623, 7.898542612646149894491172648043, 8.640069145007169692386781121596
