| L(s) = 1 | + 5.41·2-s − 5.03·3-s + 21.3·4-s + 5.67·5-s − 27.2·6-s + 7·7-s + 72.1·8-s − 1.66·9-s + 30.7·10-s + 11·11-s − 107.·12-s − 77.6·13-s + 37.9·14-s − 28.5·15-s + 220.·16-s + 45.4·17-s − 9.02·18-s − 71.6·19-s + 120.·20-s − 35.2·21-s + 59.5·22-s − 140.·23-s − 363.·24-s − 92.8·25-s − 420.·26-s + 144.·27-s + 149.·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.968·3-s + 2.66·4-s + 0.507·5-s − 1.85·6-s + 0.377·7-s + 3.18·8-s − 0.0616·9-s + 0.971·10-s + 0.301·11-s − 2.58·12-s − 1.65·13-s + 0.723·14-s − 0.491·15-s + 3.44·16-s + 0.648·17-s − 0.118·18-s − 0.865·19-s + 1.35·20-s − 0.366·21-s + 0.577·22-s − 1.27·23-s − 3.08·24-s − 0.742·25-s − 3.17·26-s + 1.02·27-s + 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.436827508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.436827508\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 5.41T + 8T^{2} \) |
| 3 | \( 1 + 5.03T + 27T^{2} \) |
| 5 | \( 1 - 5.67T + 125T^{2} \) |
| 13 | \( 1 + 77.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.75T + 2.43e4T^{2} \) |
| 31 | \( 1 + 67.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 65.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 351.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 695.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 887.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 130.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 236.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 428.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 418.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 533.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 630.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 561.T + 9.12e5T^{2} \) |
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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
−14.18119565959573716620697195151, −12.72002673040501049564559395636, −12.08364680702540331628114876278, −11.22474701907120529317633291629, −10.08421800482448471668803541506, −7.52973038249746534999327560179, −6.19071871044673133053573434768, −5.43898328882929504099683199673, −4.30681468581577511200178943350, −2.32316851763944999091484131038,
2.32316851763944999091484131038, 4.30681468581577511200178943350, 5.43898328882929504099683199673, 6.19071871044673133053573434768, 7.52973038249746534999327560179, 10.08421800482448471668803541506, 11.22474701907120529317633291629, 12.08364680702540331628114876278, 12.72002673040501049564559395636, 14.18119565959573716620697195151
