| L(s) = 1 | + (2.98 − 0.323i)3-s + 2.23i·5-s + 4.72·7-s + (8.79 − 1.92i)9-s + 4.76i·11-s − 1.06·13-s + (0.722 + 6.66i)15-s − 26.7i·17-s − 8.12·19-s + (14.0 − 1.52i)21-s + 40.0i·23-s − 5.00·25-s + (25.5 − 8.59i)27-s − 20.8i·29-s − 33.7·31-s + ⋯ |
| L(s) = 1 | + (0.994 − 0.107i)3-s + 0.447i·5-s + 0.675·7-s + (0.976 − 0.214i)9-s + 0.433i·11-s − 0.0820·13-s + (0.0481 + 0.444i)15-s − 1.57i·17-s − 0.427·19-s + (0.671 − 0.0727i)21-s + 1.74i·23-s − 0.200·25-s + (0.948 − 0.318i)27-s − 0.719i·29-s − 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.93676 + 0.104628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93676 + 0.104628i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.98 + 0.323i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 - 4.72T + 49T^{2} \) |
| 11 | \( 1 - 4.76iT - 121T^{2} \) |
| 13 | \( 1 + 1.06T + 169T^{2} \) |
| 17 | \( 1 + 26.7iT - 289T^{2} \) |
| 19 | \( 1 + 8.12T + 361T^{2} \) |
| 23 | \( 1 - 40.0iT - 529T^{2} \) |
| 29 | \( 1 + 20.8iT - 841T^{2} \) |
| 31 | \( 1 + 33.7T + 961T^{2} \) |
| 37 | \( 1 + 60.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 59.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 56.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 9.68iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 93.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 57.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 65.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 15.2T + 9.40e3T^{2} \) |
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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
−13.58667780115426225706777985185, −12.26673439813110570603616155149, −11.20241520511278772220741003096, −9.920352388378975035336751090998, −9.020725154562645573311222347825, −7.74108659774041396643565753725, −7.00716539351384249907739291293, −5.10010757774716832717239060328, −3.55885438397420757987690286948, −2.02205308251388566271748071326,
1.84567762268849174823633332001, 3.67969054727501125184306429972, 4.96969156684280485027445195783, 6.69940941079789678308585214392, 8.316459107559267785222882087077, 8.536722432636143030968281733798, 10.02783160736257436601143860622, 10.99453288307351358584877952380, 12.49694424069846902672534332772, 13.18015403583627303152618400261
