| L(s) = 1 | − i·2-s − 4-s + (0.222 + 0.222i)5-s + (−0.707 + 0.707i)7-s + i·8-s + (0.222 − 0.222i)10-s + (−1.72 + 1.72i)11-s − 1.09·13-s + (0.707 + 0.707i)14-s + 16-s + (3.72 + 1.77i)17-s − 2.44i·19-s + (−0.222 − 0.222i)20-s + (1.72 + 1.72i)22-s + (−2.05 + 2.05i)23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.0993 + 0.0993i)5-s + (−0.267 + 0.267i)7-s + 0.353i·8-s + (0.0702 − 0.0702i)10-s + (−0.521 + 0.521i)11-s − 0.305·13-s + (0.188 + 0.188i)14-s + 0.250·16-s + (0.902 + 0.429i)17-s − 0.560i·19-s + (−0.0496 − 0.0496i)20-s + (0.368 + 0.368i)22-s + (−0.427 + 0.427i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2582174334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2582174334\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 + (-3.72 - 1.77i)T \) |
| good | 5 | \( 1 + (-0.222 - 0.222i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.72 - 1.72i)T - 11iT^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + (2.05 - 2.05i)T - 23iT^{2} \) |
| 29 | \( 1 + (2.72 + 2.72i)T + 29iT^{2} \) |
| 31 | \( 1 + (4 + 4i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.12 + 3.12i)T + 37iT^{2} \) |
| 41 | \( 1 + (5.95 - 5.95i)T - 41iT^{2} \) |
| 43 | \( 1 + 11.5iT - 43T^{2} \) |
| 47 | \( 1 + 3.45T + 47T^{2} \) |
| 53 | \( 1 - 12.3iT - 53T^{2} \) |
| 59 | \( 1 + 7.54iT - 59T^{2} \) |
| 61 | \( 1 + (10.3 - 10.3i)T - 61iT^{2} \) |
| 67 | \( 1 - 6.54T + 67T^{2} \) |
| 71 | \( 1 + (6.29 + 6.29i)T + 71iT^{2} \) |
| 73 | \( 1 + (6.95 + 6.95i)T + 73iT^{2} \) |
| 79 | \( 1 + (-1.55 + 1.55i)T - 79iT^{2} \) |
| 83 | \( 1 - 0.455iT - 83T^{2} \) |
| 89 | \( 1 - 1.34T + 89T^{2} \) |
| 97 | \( 1 + (8.30 + 8.30i)T + 97iT^{2} \) |
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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
−8.836223988565552855065691974829, −7.904054028705665061201880689947, −7.27288935069656827935692392367, −6.12533740070991945827829677216, −5.41276485602597671533864698273, −4.49439208618125568345987105924, −3.56313472568266439625327169225, −2.61368604928706185863934941279, −1.73666434860583052981097063000, −0.088700047692839213566935500585,
1.47184512759649175035051999195, 3.04264254398743246665236395388, 3.77100436816413782134491872827, 5.03165137736807331294077639011, 5.47920628008083983463618751682, 6.43460286030992938637125228326, 7.20968061295944605231960242835, 7.898689281662798816190866863901, 8.592277545075767152240276903735, 9.470633277034610053583754622858
