L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s + 1.69i·7-s − i·8-s + 9-s − 10-s + 4.55i·11-s + 12-s − 1.69·14-s − i·15-s + 16-s − 2.35·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s + 0.639i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.37i·11-s + 0.288·12-s − 0.452·14-s − 0.258i·15-s + 0.250·16-s − 0.571·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8160912384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8160912384\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.69iT - 7T^{2} \) |
| 11 | \( 1 - 4.55iT - 11T^{2} \) |
| 17 | \( 1 + 2.35T + 17T^{2} \) |
| 19 | \( 1 + 6.51iT - 19T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 - 9.07T + 29T^{2} \) |
| 31 | \( 1 + 10.6iT - 31T^{2} \) |
| 37 | \( 1 + 6.18iT - 37T^{2} \) |
| 41 | \( 1 - 3.00iT - 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 - 4.28iT - 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 4.32iT - 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 3.24iT - 67T^{2} \) |
| 71 | \( 1 + 14.9iT - 71T^{2} \) |
| 73 | \( 1 - 6.72iT - 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 7.71iT - 83T^{2} \) |
| 89 | \( 1 - 9.12iT - 89T^{2} \) |
| 97 | \( 1 + 4.40iT - 97T^{2} \) |
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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
−7.87422096569913188648727440239, −7.57282565500440446796273367690, −6.52774595925722188672868640611, −6.32016742724110061705281405754, −5.41753645083922910864165038063, −4.53837500518646834613308368130, −4.18217783411873223872517751340, −2.70181368618721728365467700752, −1.98485520790170391133114084274, −0.30340717774835447534491289212,
0.865356396450850205167519087393, 1.64974660127367491576258836081, 2.93487918029643805934258069774, 3.81180721996032855076429499071, 4.37681843133250048585484199756, 5.28252712175064401627976964208, 6.00581419016006499277032572703, 6.59321111774236030356562224266, 7.71067149517681271293787408442, 8.373155921936392215865095126674