| L(s) = 1 | + (1.82 + 2.38i)3-s − 7.37i·5-s − 2.64·7-s + (−2.35 + 8.68i)9-s + 2.61i·11-s + 6.35·13-s + (17.5 − 13.4i)15-s − 12.1i·17-s + 10.2·19-s + (−4.82 − 6.30i)21-s + 4.30i·23-s − 29.4·25-s + (−24.9 + 10.2i)27-s − 17.3i·29-s + 39.2·31-s + ⋯ |
| L(s) = 1 | + (0.607 + 0.794i)3-s − 1.47i·5-s − 0.377·7-s + (−0.261 + 0.965i)9-s + 0.237i·11-s + 0.488·13-s + (1.17 − 0.896i)15-s − 0.714i·17-s + 0.538·19-s + (−0.229 − 0.300i)21-s + 0.187i·23-s − 1.17·25-s + (−0.925 + 0.378i)27-s − 0.599i·29-s + 1.26·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.131753788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131753788\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.82 - 2.38i)T \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 + 7.37iT - 25T^{2} \) |
| 11 | \( 1 - 2.61iT - 121T^{2} \) |
| 13 | \( 1 - 6.35T + 169T^{2} \) |
| 17 | \( 1 + 12.1iT - 289T^{2} \) |
| 19 | \( 1 - 10.2T + 361T^{2} \) |
| 23 | \( 1 - 4.30iT - 529T^{2} \) |
| 29 | \( 1 + 17.3iT - 841T^{2} \) |
| 31 | \( 1 - 39.2T + 961T^{2} \) |
| 37 | \( 1 + 41.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 39.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 105. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 41.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 20.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 27.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 67.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 63.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 89.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 63.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.1T + 9.40e3T^{2} \) |
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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
−9.380239895150428251088769074280, −8.560727670463833711987528356186, −8.054504015276714280863227667146, −6.95251275277709673688985145568, −5.63434940554510741316319170455, −4.99618446851153533571083061154, −4.20179064838208547084971939518, −3.29933081880371573659819282622, −2.02658835640111546943122020019, −0.61063241991143029314758919049,
1.20756976373146912393641328859, 2.57628558395088126864059826883, 3.14532177755826426681166259217, 4.05568677919573735909092903878, 5.78064601267665451605016038151, 6.42964763465501524379673729507, 7.05425813950295470279071168650, 7.81236169760713827289434589967, 8.622597998777590917246391404950, 9.502462185999333749327037013466
