| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 1.56i·7-s − i·8-s − 9-s − 1.56·11-s − i·12-s − 2i·13-s + 1.56·14-s + 16-s + 5.12i·17-s − i·18-s − 4.68·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.590i·7-s − 0.353i·8-s − 0.333·9-s − 0.470·11-s − 0.288i·12-s − 0.554i·13-s + 0.417·14-s + 0.250·16-s + 1.24i·17-s − 0.235i·18-s − 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.429001531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.429001531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.56iT - 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 5.12iT - 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 + 5.56iT - 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 37 | \( 1 - 5.12iT - 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.80iT - 43T^{2} \) |
| 47 | \( 1 + 3.12iT - 47T^{2} \) |
| 53 | \( 1 + 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 4.87T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 9.36iT - 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 9.80iT - 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 2.24iT - 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.237833960591208022785014882874, −7.984165673191744371984755480426, −6.81047621081432969713266473775, −6.36047602896710365436997528561, −5.51142217494776852550288840647, −4.72189350679260254605982719829, −4.10570864890589416960167320056, −3.30929073167262113917873258344, −2.16597438457862505212061249361, −0.63357346765610082826194222411,
0.66502963529881758785955748632, 1.96165774967665834184026430781, 2.48778996783538632993947573265, 3.43426961638120682080913853913, 4.39734108172500751164141073311, 5.25549167855958251282433018805, 5.86849769472132214116281617817, 6.83213845253569597333418431199, 7.48310681114126365531367480313, 8.273700842259963172022143000652
