L(s) = 1 | + (1.34 − 0.444i)2-s + (1.29 + 2.24i)3-s + (1.60 − 1.19i)4-s + (2.74 + 2.44i)6-s + (−0.603 + 2.57i)7-s + (1.62 − 2.31i)8-s + (−1.87 + 3.23i)9-s + (3.12 − 1.80i)11-s + (4.76 + 2.05i)12-s + 0.818i·13-s + (0.335 + 3.72i)14-s + (1.14 − 3.83i)16-s + (−6.40 + 3.69i)17-s + (−1.06 + 5.18i)18-s + (1.65 − 2.86i)19-s + ⋯ |
L(s) = 1 | + (0.949 − 0.314i)2-s + (0.749 + 1.29i)3-s + (0.801 − 0.597i)4-s + (1.11 + 0.996i)6-s + (−0.228 + 0.973i)7-s + (0.573 − 0.819i)8-s + (−0.623 + 1.07i)9-s + (0.940 − 0.543i)11-s + (1.37 + 0.593i)12-s + 0.226i·13-s + (0.0896 + 0.995i)14-s + (0.286 − 0.958i)16-s + (−1.55 + 0.896i)17-s + (−0.251 + 1.22i)18-s + (0.379 − 0.656i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.10677 + 1.32585i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.10677 + 1.32585i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.34 + 0.444i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.603 - 2.57i)T \) |
good | 3 | \( 1 + (-1.29 - 2.24i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.818iT - 13T^{2} \) |
| 17 | \( 1 + (6.40 - 3.69i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.65 + 2.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.19 + 1.26i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 + (0.955 + 1.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.58 + 6.20i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.65iT - 41T^{2} \) |
| 43 | \( 1 - 2.39iT - 43T^{2} \) |
| 47 | \( 1 + (0.667 - 1.15i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.905 + 1.56i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.955 + 1.65i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.46 + 4.88i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.02 + 4.63i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.38iT - 71T^{2} \) |
| 73 | \( 1 + (6.40 - 3.69i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.70 + 3.87i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + (-9.19 - 5.30i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.32iT - 97T^{2} \) |
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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
−10.73859731080810277540915775230, −9.570642050293273307733663930609, −9.149391952036032002497035857771, −8.262358649330011678059044976252, −6.65819902292142555744966803323, −5.93654928309815464224138649957, −4.74500408864086426672556119606, −4.05723101539859471698791426313, −3.14274936962135782707976750528, −2.14012010512873123268824803586,
1.46418381017547992370278153656, 2.62699489942454114093844256247, 3.75208078125775837728523578632, 4.70980034197263370546765609297, 6.22769842144506567610807645568, 6.89269749748130603336045688138, 7.40500661269961174082554587780, 8.244828824549806092126366250648, 9.271430056317517264903406728500, 10.48413074360443783787350819027