L(s) = 1 | + 2.23·2-s + (2.23 + 2i)3-s + 1.00·4-s + (5.00 + 4.47i)6-s − 6i·7-s − 6.70·8-s + (1.00 + 8.94i)9-s + 4.47i·11-s + (2.23 + 2.00i)12-s − 16i·13-s − 13.4i·14-s − 19·16-s − 4.47·17-s + (2.23 + 20.0i)18-s + 2·19-s + ⋯ |
L(s) = 1 | + 1.11·2-s + (0.745 + 0.666i)3-s + 0.250·4-s + (0.833 + 0.745i)6-s − 0.857i·7-s − 0.838·8-s + (0.111 + 0.993i)9-s + 0.406i·11-s + (0.186 + 0.166i)12-s − 1.23i·13-s − 0.958i·14-s − 1.18·16-s − 0.263·17-s + (0.124 + 1.11i)18-s + 0.105·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.22757 + 0.425429i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.22757 + 0.425429i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.23 - 2i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2.23T + 4T^{2} \) |
| 7 | \( 1 + 6iT - 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 + 16iT - 169T^{2} \) |
| 17 | \( 1 + 4.47T + 289T^{2} \) |
| 19 | \( 1 - 2T + 361T^{2} \) |
| 23 | \( 1 - 13.4T + 529T^{2} \) |
| 29 | \( 1 - 31.3iT - 841T^{2} \) |
| 31 | \( 1 + 18T + 961T^{2} \) |
| 37 | \( 1 + 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 16iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 49.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 4.47T + 2.80e3T^{2} \) |
| 59 | \( 1 + 4.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82T + 3.72e3T^{2} \) |
| 67 | \( 1 - 24iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 138T + 6.24e3T^{2} \) |
| 83 | \( 1 + 93.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 166iT - 9.40e3T^{2} \) |
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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
−14.37068391742290327613419582952, −13.37829718466998463341785714974, −12.68943509124999069698359647709, −11.02757305754365312200304710091, −9.957489967599238255946509188327, −8.695916637809971321562132067150, −7.27020219943647043171970581362, −5.39194695345011453929921597559, −4.24628357171964507243442624048, −3.07297585787477368925610568184,
2.48672149529654840341633430622, 4.00876422713434947061335680192, 5.69209964193781473695484719826, 6.88725113423520709060004447181, 8.594793722974099512644820849458, 9.347234988067112550107610244410, 11.53350113638795515282405894301, 12.31657548681470100757168704680, 13.32414707109295385458927180571, 14.06902651332128290111432432242