| L(s) = 1 | − 1.65i·2-s + 1.97i·3-s − 0.737·4-s + (2.19 + 0.434i)5-s + 3.26·6-s + 2.24i·7-s − 2.08i·8-s − 0.899·9-s + (0.718 − 3.62i)10-s − 1.45i·12-s − 3.69i·13-s + 3.71·14-s + (−0.857 + 4.33i)15-s − 4.93·16-s + 2.22i·17-s + 1.48i·18-s + ⋯ |
| L(s) = 1 | − 1.16i·2-s + 1.14i·3-s − 0.368·4-s + (0.980 + 0.194i)5-s + 1.33·6-s + 0.847i·7-s − 0.738i·8-s − 0.299·9-s + (0.227 − 1.14i)10-s − 0.420i·12-s − 1.02i·13-s + 0.991·14-s + (−0.221 + 1.11i)15-s − 1.23·16-s + 0.539i·17-s + 0.350i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.90893 - 0.187073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90893 - 0.187073i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.19 - 0.434i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.65iT - 2T^{2} \) |
| 3 | \( 1 - 1.97iT - 3T^{2} \) |
| 7 | \( 1 - 2.24iT - 7T^{2} \) |
| 13 | \( 1 + 3.69iT - 13T^{2} \) |
| 17 | \( 1 - 2.22iT - 17T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 - 3.85iT - 23T^{2} \) |
| 29 | \( 1 - 0.188T + 29T^{2} \) |
| 31 | \( 1 - 0.686T + 31T^{2} \) |
| 37 | \( 1 - 2.59iT - 37T^{2} \) |
| 41 | \( 1 + 7.91T + 41T^{2} \) |
| 43 | \( 1 - 8.41iT - 43T^{2} \) |
| 47 | \( 1 + 12.0iT - 47T^{2} \) |
| 53 | \( 1 + 12.6iT - 53T^{2} \) |
| 59 | \( 1 - 0.343T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 - 0.650iT - 67T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 + 8.85iT - 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 - 3.18iT - 83T^{2} \) |
| 89 | \( 1 + 9.92T + 89T^{2} \) |
| 97 | \( 1 - 2.26iT - 97T^{2} \) |
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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
−10.48307017985153470817163316922, −9.862472146030472498232243370224, −9.494534942428119802582891474236, −8.425832648146911590800071797825, −6.93576144195954951316192582057, −5.70365554576105478895747441894, −5.01647258311683613008306677728, −3.57963331110942288720055787348, −2.88616389014417602352742694798, −1.60472329025461557159818716344,
1.28775756720930651115278415518, 2.50101592128793138426813582401, 4.47771390923837335977357044692, 5.54883788246776799667525313383, 6.47392282108887510881449553182, 7.03578262515533199134353088118, 7.61859382225143485193451457438, 8.686885390881265781388094972419, 9.565786218733175026780758390523, 10.62638322124869894926166675908
