L(s) = 1 | + (1.39 − 0.207i)2-s + 1.47i·3-s + (1.91 − 0.579i)4-s + (0.305 + 2.06i)6-s + (−0.819 + 2.51i)7-s + (2.55 − 1.20i)8-s + 0.828·9-s + 2.79i·11-s + (0.853 + 2.82i)12-s − 5.83·13-s + (−0.625 + 3.68i)14-s + (3.32 − 2.21i)16-s + 4.12·17-s + (1.15 − 0.171i)18-s + 5.64·19-s + ⋯ |
L(s) = 1 | + (0.989 − 0.146i)2-s + 0.850i·3-s + (0.957 − 0.289i)4-s + (0.124 + 0.841i)6-s + (−0.309 + 0.950i)7-s + (0.904 − 0.426i)8-s + 0.276·9-s + 0.843i·11-s + (0.246 + 0.814i)12-s − 1.61·13-s + (−0.167 + 0.985i)14-s + (0.832 − 0.554i)16-s + 0.999·17-s + (0.273 − 0.0404i)18-s + 1.29·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.38175 + 1.43771i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.38175 + 1.43771i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 + 0.207i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.819 - 2.51i)T \) |
good | 3 | \( 1 - 1.47iT - 3T^{2} \) |
| 11 | \( 1 - 2.79iT - 11T^{2} \) |
| 13 | \( 1 + 5.83T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 - 0.242T + 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 + 6.24iT - 37T^{2} \) |
| 41 | \( 1 + 4.12iT - 41T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 + 6.25iT - 47T^{2} \) |
| 53 | \( 1 - 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 2.94T + 59T^{2} \) |
| 61 | \( 1 + 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 7.43T + 67T^{2} \) |
| 71 | \( 1 + 7.23iT - 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 7.91iT - 79T^{2} \) |
| 83 | \( 1 - 1.47iT - 83T^{2} \) |
| 89 | \( 1 + 12.3iT - 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−10.49029991750555403264526131042, −9.794549153060273143233897558246, −9.372523172075278859977103700464, −7.66794631836289873998431705104, −7.07807579143394954010456502571, −5.66984622611191455290654025678, −5.14057143776003852881295650796, −4.20466202196102214528873907998, −3.13505712984721936705027407276, −2.04626337067381079835487379965,
1.17538802946832541639135451976, 2.68880666049150549261804401125, 3.72210718877009363891774466550, 4.85082106395993537378852247995, 5.85288952868668971410462897858, 6.82878766562223915364395444821, 7.50199763922747728825312904609, 7.977014332433320677259852136226, 9.751082272221029144230608230177, 10.29863470305571557139097256628