| L(s) = 1 | − 4.09e3i·2-s − 1.24e6i·3-s − 1.67e7·4-s − 5.10e9·6-s + 3.92e10i·7-s + 6.87e10i·8-s − 7.08e11·9-s − 1.50e13·11-s + 2.09e13i·12-s + 1.45e14i·13-s + 1.60e14·14-s + 2.81e14·16-s + 1.30e15i·17-s + 2.90e15i·18-s + 9.75e15·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.35i·3-s − 0.5·4-s − 0.958·6-s + 1.07i·7-s + 0.353i·8-s − 0.835·9-s − 1.44·11-s + 0.677i·12-s + 1.73i·13-s + 0.758·14-s + 0.250·16-s + 0.542i·17-s + 0.590i·18-s + 1.01·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.5237523532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5237523532\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.24e6iT - 8.47e11T^{2} \) |
| 7 | \( 1 - 3.92e10iT - 1.34e21T^{2} \) |
| 11 | \( 1 + 1.50e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.45e14iT - 7.05e27T^{2} \) |
| 17 | \( 1 - 1.30e15iT - 5.77e30T^{2} \) |
| 19 | \( 1 - 9.75e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.00e17iT - 1.10e34T^{2} \) |
| 29 | \( 1 - 1.41e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 3.30e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 6.55e19iT - 1.60e39T^{2} \) |
| 41 | \( 1 + 9.42e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.20e20iT - 6.86e40T^{2} \) |
| 47 | \( 1 + 8.86e20iT - 6.34e41T^{2} \) |
| 53 | \( 1 - 6.79e21iT - 1.27e43T^{2} \) |
| 59 | \( 1 - 1.11e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 3.14e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 4.52e22iT - 4.48e45T^{2} \) |
| 71 | \( 1 + 3.31e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.04e23iT - 3.82e46T^{2} \) |
| 79 | \( 1 - 1.45e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 8.80e23iT - 9.48e47T^{2} \) |
| 89 | \( 1 - 1.79e23T + 5.42e48T^{2} \) |
| 97 | \( 1 + 7.02e24iT - 4.66e49T^{2} \) |
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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
−10.37409502634966465605350891523, −9.011323254333207781823553708566, −8.121187747014433232612402238733, −6.95679904114462605371394553445, −5.83591937237979454060190926086, −4.65330664341459574898701971695, −2.94042150700405804658016344491, −2.14999676280918708431829459202, −1.41903913306791764844112626345, −0.11904702918877395204527773977,
0.816632847912762244215606200004, 2.97046463722268900904399722620, 3.78407562616293340657949270870, 5.04569103556410237178510970431, 5.48265172401808364414376458014, 7.32374066932254955689060958180, 8.016841083597146001256266820368, 9.511601016338520105455151416104, 10.26864429801513532730816861620, 10.95874986537718528805672161297
