| L(s) = 1 | + (0.417 + 2.63i)2-s + (0.870 − 1.49i)3-s + (−4.86 + 1.57i)4-s + (3.09 + 1.57i)5-s + (4.30 + 1.66i)6-s + (−1.40 + 0.336i)7-s + (−3.76 − 7.39i)8-s + (−1.48 − 2.60i)9-s + (−2.86 + 8.81i)10-s + (−0.761 + 0.891i)11-s + (−1.86 + 8.65i)12-s + (−1.57 − 2.57i)13-s + (−1.46 − 3.54i)14-s + (5.06 − 3.26i)15-s + (9.62 − 6.99i)16-s + (1.40 − 0.110i)17-s + ⋯ |
| L(s) = 1 | + (0.294 + 1.86i)2-s + (0.502 − 0.864i)3-s + (−2.43 + 0.789i)4-s + (1.38 + 0.705i)5-s + (1.75 + 0.681i)6-s + (−0.529 + 0.127i)7-s + (−1.33 − 2.61i)8-s + (−0.494 − 0.869i)9-s + (−0.906 + 2.78i)10-s + (−0.229 + 0.268i)11-s + (−0.539 + 2.49i)12-s + (−0.437 − 0.713i)13-s + (−0.392 − 0.948i)14-s + (1.30 − 0.842i)15-s + (2.40 − 1.74i)16-s + (0.339 − 0.0267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.783276 + 1.10122i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.783276 + 1.10122i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.870 + 1.49i)T \) |
| 41 | \( 1 + (6.01 - 2.19i)T \) |
| good | 2 | \( 1 + (-0.417 - 2.63i)T + (-1.90 + 0.618i)T^{2} \) |
| 5 | \( 1 + (-3.09 - 1.57i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (1.40 - 0.336i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (0.761 - 0.891i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.57 + 2.57i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-1.40 + 0.110i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-1.83 + 2.98i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (0.944 + 0.686i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-8.46 - 0.666i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (6.63 + 2.15i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.91 - 5.89i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (1.99 - 0.316i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.47 + 6.12i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (0.150 - 1.91i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (7.54 - 10.3i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.823 + 5.20i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.115 + 0.135i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-2.47 - 2.11i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-2.36 + 2.36i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.24 + 3.00i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 5.05iT - 83T^{2} \) |
| 89 | \( 1 + (-1.90 - 7.91i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (9.56 - 8.16i)T + (15.1 - 95.8i)T^{2} \) |
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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
−13.80434250006906571940341354393, −13.33886864765248790867767122675, −12.40632765617908633147225940632, −10.04521704917163452486431696089, −9.204945658194529766180844507501, −7.996051614336710474237957931146, −6.94098494371683146734534475970, −6.28533146681360506375845414151, −5.27006942873000036480598131791, −2.97713902631352727834208155778,
1.93144418982544044880967561485, 3.28795550241126368353792993045, 4.66942644678350251127372180333, 5.65664185716807108791626562173, 8.572307246781128093591746859935, 9.535178186139283572947825379091, 9.894869332830378346908620316046, 10.83400553755866284310706363834, 12.15977934468988933761320966764, 13.05268887742233711786990819710
