| L(s) = 1 | − 2i·2-s + 3·3-s − 4·4-s − 10.4i·5-s − 6i·6-s + 18.5i·7-s + 8i·8-s + 9·9-s − 20.8·10-s + 18.2i·11-s − 12·12-s + 37.0·14-s − 31.2i·15-s + 16·16-s + 20.7·17-s − 18i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.931i·5-s − 0.408i·6-s + 0.999i·7-s + 0.353i·8-s + 0.333·9-s − 0.658·10-s + 0.499i·11-s − 0.288·12-s + 0.706·14-s − 0.537i·15-s + 0.250·16-s + 0.296·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.278059611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.278059611\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 10.4iT - 125T^{2} \) |
| 7 | \( 1 - 18.5iT - 343T^{2} \) |
| 11 | \( 1 - 18.2iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 20.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 84.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 87.7iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 323. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 206. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 382. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 284.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 715. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 759.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.08e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 938. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 308. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 15.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.48e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 883. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 446. iT - 9.12e5T^{2} \) |
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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
−9.515588966840670125974220546517, −8.761341142175230273588527958318, −8.300138049763046020101062160789, −7.20218428385245624087545641200, −5.89037396835810952756425834342, −4.98487453915853263414091578948, −4.21980748098575527996516709234, −2.97896407058695400155332425860, −2.10497088930229016082303239056, −1.00302215110448648613523595286,
0.63052218825624962829604378108, 2.24027567721380744803911601249, 3.58730595410562237490578002103, 4.03176817018017787930828035013, 5.49372708014621013852046769946, 6.34674128256652751940066668714, 7.32200941253185218756104713569, 7.64303018231245212830916921859, 8.649873687845723435526746067085, 9.544900095828233029886773035590
