| L(s) = 1 | + (0.275 + 4.99i)5-s + (6.91 + 1.06i)7-s − 3.07·11-s − 17.8·13-s − 24.0·17-s − 32.7i·19-s − 24.3i·23-s + (−24.8 + 2.75i)25-s + 18.6·29-s − 44.2i·31-s + (−3.41 + 34.8i)35-s − 37.2i·37-s + 57.4i·41-s − 59.9i·43-s + 17.6·47-s + ⋯ |
| L(s) = 1 | + (0.0551 + 0.998i)5-s + (0.988 + 0.152i)7-s − 0.279·11-s − 1.37·13-s − 1.41·17-s − 1.72i·19-s − 1.05i·23-s + (−0.993 + 0.110i)25-s + 0.643·29-s − 1.42i·31-s + (−0.0975 + 0.995i)35-s − 1.00i·37-s + 1.40i·41-s − 1.39i·43-s + 0.375·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0975 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0975 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9575168612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9575168612\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.275 - 4.99i)T \) |
| 7 | \( 1 + (-6.91 - 1.06i)T \) |
| good | 11 | \( 1 + 3.07T + 121T^{2} \) |
| 13 | \( 1 + 17.8T + 169T^{2} \) |
| 17 | \( 1 + 24.0T + 289T^{2} \) |
| 19 | \( 1 + 32.7iT - 361T^{2} \) |
| 23 | \( 1 + 24.3iT - 529T^{2} \) |
| 29 | \( 1 - 18.6T + 841T^{2} \) |
| 31 | \( 1 + 44.2iT - 961T^{2} \) |
| 37 | \( 1 + 37.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 57.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 59.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 17.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 18.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 85.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 75.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 29.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 47.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 11.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 40.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 26.3T + 9.40e3T^{2} \) |
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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
−9.251625334087354347277415483196, −8.494790935914786234784491979165, −7.43637176043149297389942163095, −7.00795892446615576057684790878, −6.00535007661889815490417897870, −4.86190713773910469403095218243, −4.29470373254312276128742697435, −2.58738490189120703921807043107, −2.31579478767047888740022753538, −0.26960936174296409557023440358,
1.33147827081022514662250590622, 2.24325591205646676407046441665, 3.78675583161852236044439074282, 4.84958639389147828657701017134, 5.15338748579001722923290406502, 6.36443775833622754609530809511, 7.50285524988099183748198766384, 8.084872395791085020472689998266, 8.827328638778671583999237129914, 9.686226078119559601484220208926
