| L(s) = 1 | − 7.04i·5-s − 3.07·7-s + 7.06i·11-s − 21.0·13-s − 2.77i·17-s − 4.35·19-s + 18.0i·23-s − 24.5·25-s + 19.2i·29-s + 7.72·31-s + 21.6i·35-s + 7.60·37-s + 36.9i·41-s + 45.9·43-s + 58.6i·47-s + ⋯ |
| L(s) = 1 | − 1.40i·5-s − 0.439·7-s + 0.642i·11-s − 1.61·13-s − 0.163i·17-s − 0.229·19-s + 0.785i·23-s − 0.983·25-s + 0.663i·29-s + 0.249·31-s + 0.619i·35-s + 0.205·37-s + 0.901i·41-s + 1.06·43-s + 1.24i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2548859293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2548859293\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35T \) |
| good | 5 | \( 1 + 7.04iT - 25T^{2} \) |
| 7 | \( 1 + 3.07T + 49T^{2} \) |
| 11 | \( 1 - 7.06iT - 121T^{2} \) |
| 13 | \( 1 + 21.0T + 169T^{2} \) |
| 17 | \( 1 + 2.77iT - 289T^{2} \) |
| 23 | \( 1 - 18.0iT - 529T^{2} \) |
| 29 | \( 1 - 19.2iT - 841T^{2} \) |
| 31 | \( 1 - 7.72T + 961T^{2} \) |
| 37 | \( 1 - 7.60T + 1.36e3T^{2} \) |
| 41 | \( 1 - 36.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 45.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 23.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 72.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 86.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 59.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 42.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 30.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 41.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 67.6T + 9.40e3T^{2} \) |
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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
−10.38204758045499047206996570691, −9.474197159047321915113178866404, −9.137178306583665353335452499576, −7.914818112263549703559585298734, −7.24564856246208856076417101866, −5.99372993616017772145034709613, −4.92872833429092294269493933173, −4.41946176434106755913596854378, −2.84502116866579817883467014236, −1.44752628328879239752694722921,
0.088470561905602449401428179451, 2.35269864540260126078627991704, 3.08144792923977352527011556521, 4.28301413283964657379866591906, 5.61323144830275842364509700362, 6.56342809199362826110204888825, 7.18681490234394158865884002057, 8.107773789522100903098368064484, 9.282305303720476180708846287445, 10.15597626832320573108528598086
