| L(s) = 1 | − 2i·2-s − 6.14i·3-s − 4·4-s − 12.2·6-s − 22.2i·7-s + 8i·8-s − 10.7·9-s + 43.8·11-s + 24.5i·12-s + 13i·13-s − 44.5·14-s + 16·16-s − 99.9i·17-s + 21.4i·18-s − 93.2·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.18i·3-s − 0.5·4-s − 0.835·6-s − 1.20i·7-s + 0.353i·8-s − 0.396·9-s + 1.20·11-s + 0.590i·12-s + 0.277i·13-s − 0.850·14-s + 0.250·16-s − 1.42i·17-s + 0.280i·18-s − 1.12·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.387507098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387507098\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 13iT \) |
| good | 3 | \( 1 + 6.14iT - 27T^{2} \) |
| 7 | \( 1 + 22.2iT - 343T^{2} \) |
| 11 | \( 1 - 43.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 99.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 154. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 401. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 458.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 251. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 261. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 151. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 263.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 387. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 544.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 301. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 562.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 938. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3iT - 9.12e5T^{2} \) |
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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
−9.583134750783927182731931162000, −8.783064540855245872261815101393, −7.63564410740955374899030179025, −6.98148891071440943876842982072, −6.23450185824611668639685535895, −4.61342032717304698235071367362, −3.83871973484181012606047081883, −2.39265691802691178142350942157, −1.30887164959187708916911587928, −0.41783268480622967312699676645,
1.86005630512049669765424506836, 3.65859151428014854921268074816, 4.17164660630640255851015875600, 5.54444899720194462599125002847, 5.91560385242636328222508725378, 7.18125147808258148951357098536, 8.343281363618609285283165272451, 9.208549362300445401806837737369, 9.439144960605225633287249505673, 10.66961331563339887486717256018
