L(s) = 1 | − 2.08i·2-s + 2.19i·3-s − 2.34·4-s + 4.58·6-s + 0.992i·7-s + 0.726i·8-s − 1.83·9-s + 2·11-s − 5.16i·12-s + 3.37i·13-s + 2.06·14-s − 3.18·16-s + 2.89i·17-s + 3.82i·18-s + 2.58·19-s + ⋯ |
L(s) = 1 | − 1.47i·2-s + 1.26i·3-s − 1.17·4-s + 1.87·6-s + 0.375i·7-s + 0.256i·8-s − 0.611·9-s + 0.603·11-s − 1.49i·12-s + 0.935i·13-s + 0.553·14-s − 0.795·16-s + 0.702i·17-s + 0.901i·18-s + 0.592·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44828\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + 2.08iT - 2T^{2} \) |
| 3 | \( 1 - 2.19iT - 3T^{2} \) |
| 7 | \( 1 - 0.992iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 - 2.89iT - 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 4.54iT - 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 0.136T + 31T^{2} \) |
| 37 | \( 1 + 2.14iT - 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 + 4.64iT - 43T^{2} \) |
| 47 | \( 1 - 9.92iT - 47T^{2} \) |
| 53 | \( 1 + 7.56iT - 53T^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 2.18iT - 67T^{2} \) |
| 71 | \( 1 - 9.64T + 71T^{2} \) |
| 73 | \( 1 + 0.775iT - 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 1.77iT - 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 17.0iT - 97T^{2} \) |
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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
−10.71258913956373070916988698415, −9.720959733632630671937613928362, −9.421848048364219610651371341130, −8.575836866344884047546368753854, −7.03238193690150400286030751513, −5.73015471711992844409056218108, −4.48527994715935859344201098111, −3.93229807316909968410896507126, −2.92866501857315221728458968662, −1.54459962503501652282022184823,
0.891605398386285778221091692255, 2.66344085466365775473730107319, 4.40239656489755064252275269882, 5.53018465512495221968321406194, 6.41883854166076570306138838505, 7.02494590598812800785261729146, 7.72838044515040667764814832692, 8.362061383026194372221297607555, 9.371202112711464115517070381587, 10.55132491165536964485064226631