L(s) = 1 | − 0.519·5-s + 10.4·7-s + 14.1·11-s + 5.39i·13-s + 24.9i·17-s + 13.3i·19-s + 41.1i·23-s − 24.7·25-s − 7.85·29-s − 26.1·31-s − 5.40·35-s + 1.53i·37-s + 21.3i·41-s − 64.1i·43-s − 19.8i·47-s + ⋯ |
L(s) = 1 | − 0.103·5-s + 1.48·7-s + 1.28·11-s + 0.414i·13-s + 1.46i·17-s + 0.701i·19-s + 1.78i·23-s − 0.989·25-s − 0.270·29-s − 0.844·31-s − 0.154·35-s + 0.0415i·37-s + 0.520i·41-s − 1.49i·43-s − 0.422i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.312255243\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.312255243\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 0.519T + 25T^{2} \) |
| 7 | \( 1 - 10.4T + 49T^{2} \) |
| 11 | \( 1 - 14.1T + 121T^{2} \) |
| 13 | \( 1 - 5.39iT - 169T^{2} \) |
| 17 | \( 1 - 24.9iT - 289T^{2} \) |
| 19 | \( 1 - 13.3iT - 361T^{2} \) |
| 23 | \( 1 - 41.1iT - 529T^{2} \) |
| 29 | \( 1 + 7.85T + 841T^{2} \) |
| 31 | \( 1 + 26.1T + 961T^{2} \) |
| 37 | \( 1 - 1.53iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 21.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 64.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 19.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 68.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 67.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 58.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 56.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 107. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7.73T + 5.32e3T^{2} \) |
| 79 | \( 1 - 64.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 125.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 59.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−9.193533812824387376080730305516, −8.470954597284647167937094896646, −7.79068383706002292388010385623, −7.03771706807719488483613526023, −5.94550480291714913345409499762, −5.32410215524935476794501517236, −4.05217016944783085854061944407, −3.77196518466979477458574034834, −1.85369169684215421626223126886, −1.45868127219058093756529625011,
0.62413844774624386416383810769, 1.75959940004790328647624143698, 2.84440054902694372256943152682, 4.17469860338650836624247165147, 4.72642767785902498116683431152, 5.63995644414880275964171033906, 6.67512635506554320103363412675, 7.44062062944434351605533008020, 8.176899004044076282694871699113, 8.978046880870611674774989714058