| L(s) = 1 | + (−0.253 + 1.39i)2-s + (0.987 + 0.156i)3-s + (−1.87 − 0.705i)4-s + (−0.698 − 2.12i)5-s + (−0.467 + 1.33i)6-s + (1.04 − 1.04i)7-s + (1.45 − 2.42i)8-s + (0.951 + 0.309i)9-s + (3.13 − 0.433i)10-s + (4.52 − 1.47i)11-s + (−1.73 − 0.989i)12-s + (2.54 + 4.99i)13-s + (1.18 + 1.71i)14-s + (−0.357 − 2.20i)15-s + (3.00 + 2.63i)16-s + (0.262 + 1.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.179 + 0.983i)2-s + (0.570 + 0.0903i)3-s + (−0.935 − 0.352i)4-s + (−0.312 − 0.949i)5-s + (−0.191 + 0.544i)6-s + (0.394 − 0.394i)7-s + (0.514 − 0.857i)8-s + (0.317 + 0.103i)9-s + (0.990 − 0.137i)10-s + (1.36 − 0.443i)11-s + (−0.501 − 0.285i)12-s + (0.705 + 1.38i)13-s + (0.317 + 0.458i)14-s + (−0.0922 − 0.569i)15-s + (0.751 + 0.659i)16-s + (0.0635 + 0.401i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33022 + 0.335972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33022 + 0.335972i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.253 - 1.39i)T \) |
| 3 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (0.698 + 2.12i)T \) |
| good | 7 | \( 1 + (-1.04 + 1.04i)T - 7iT^{2} \) |
| 11 | \( 1 + (-4.52 + 1.47i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.54 - 4.99i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.262 - 1.65i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (2.76 + 2.01i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.62 + 7.10i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (4.76 + 6.56i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.29 - 1.77i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.10 + 0.565i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (1.73 - 5.32i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-2.44 - 2.44i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.899 + 5.67i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (0.680 - 4.29i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (3.92 - 12.0i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 6.62i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.39 + 1.17i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (7.39 + 10.1i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.08 + 4.63i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (7.49 - 5.44i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.974 - 6.15i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.92 + 0.626i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (3.36 + 0.532i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.87496208955234648646908680219, −10.80028231193995838899083555988, −9.323772787185863786607505014537, −8.895454474750803571106360370361, −8.176812428941488602898476613747, −6.99075632707529617866451500774, −6.07406630707997624819839071932, −4.42930844418341381658996793446, −4.07874092381015784790364281328, −1.31228556876323186339366216035,
1.68805007605671003256153047607, 3.15978970669630009369965829601, 3.89177803875206764537816194024, 5.51667019302152899497559457180, 7.07258279145414421306288967438, 8.034406548276619987347035162248, 8.973874091700166927702167698671, 9.840468025827566942011244653808, 10.88501411831047525113249969900, 11.49413576113093401699890791898
