| L(s) = 1 | + 2.24i·3-s − 2.23·5-s − 9.52i·7-s + 3.94·9-s + 14.5i·11-s + 7.52·13-s − 5.02i·15-s − 22.9·17-s − 10.0i·19-s + 21.4·21-s − 0.530i·23-s + 5.00·25-s + 29.1i·27-s + 19.7·29-s − 48.1i·31-s + ⋯ |
| L(s) = 1 | + 0.749i·3-s − 0.447·5-s − 1.36i·7-s + 0.438·9-s + 1.32i·11-s + 0.579·13-s − 0.335i·15-s − 1.34·17-s − 0.529i·19-s + 1.01·21-s − 0.0230i·23-s + 0.200·25-s + 1.07i·27-s + 0.681·29-s − 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.781952715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781952715\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 3 | \( 1 - 2.24iT - 9T^{2} \) |
| 7 | \( 1 + 9.52iT - 49T^{2} \) |
| 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 7.52T + 169T^{2} \) |
| 17 | \( 1 + 22.9T + 289T^{2} \) |
| 19 | \( 1 + 10.0iT - 361T^{2} \) |
| 23 | \( 1 + 0.530iT - 529T^{2} \) |
| 29 | \( 1 - 19.7T + 841T^{2} \) |
| 31 | \( 1 + 48.1iT - 961T^{2} \) |
| 37 | \( 1 + 2.58T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 78.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 73.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 61.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 12.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 7.30T + 3.72e3T^{2} \) |
| 67 | \( 1 - 42.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 106. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 58.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 62.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 18.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 123.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 46.7T + 9.40e3T^{2} \) |
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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
−9.546502960532468417612973471557, −8.867724192882640287490177214848, −7.60453431565883156707021120567, −7.20480510781748726459990848233, −6.31037792530775599070530100796, −4.77532972990563372308480654212, −4.32060415895168487453664963574, −3.73392537270458370525226060079, −2.18559119146403239847965725577, −0.70139084128785787394728032492,
0.906080178928354996119491359692, 2.15509351776435919759072531197, 3.15087634268619653115009735045, 4.29211347539251827670842066091, 5.46114880945579919986855163278, 6.26287224925685632397165167027, 6.87691912780569092823803545061, 8.062776747760943133112750033184, 8.551857820350308937076741808013, 9.171919261786435154727123958273
