L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s − 5-s − 6-s + (−0.690 + 2.12i)7-s + (−0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.466 − 1.43i)11-s + (−0.809 + 0.587i)12-s + (−5.21 − 3.79i)13-s + (0.690 + 2.12i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.837 + 2.57i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.467 − 0.339i)3-s + (0.154 − 0.475i)4-s − 0.447·5-s − 0.408·6-s + (−0.261 + 0.803i)7-s + (−0.109 − 0.336i)8-s + (0.103 + 0.317i)9-s + (−0.255 + 0.185i)10-s + (0.140 − 0.433i)11-s + (−0.233 + 0.169i)12-s + (−1.44 − 1.05i)13-s + (0.184 + 0.568i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.203 + 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00704784 + 0.0157026i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00704784 + 0.0157026i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + (5.42 + 1.27i)T \) |
good | 7 | \( 1 + (0.690 - 2.12i)T + (-5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.466 + 1.43i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (5.21 + 3.79i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.837 - 2.57i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.71 - 3.42i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.203 + 0.625i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (8.35 - 6.07i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 - 2.91T + 37T^{2} \) |
| 41 | \( 1 + (8.57 - 6.23i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.50 + 6.17i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (5.57 + 4.05i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.04 - 3.22i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (4.88 + 3.55i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 + (-3.75 - 11.5i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.27 + 7.00i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.27 - 7.01i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (5.38 - 3.91i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.35 + 4.16i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.55 + 14.0i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.56554238345581591060799123229, −9.751257199468074810771811551319, −8.679745937262696041251773490236, −7.77710496985810544214207440147, −6.85738997108629258053351989757, −5.76056676580281101379836547381, −5.33594057758159446448199456432, −4.07380794983973548041132926735, −3.02018896408247189797389528292, −1.88148623598252119048478545664,
0.00655431017867297376219462176, 2.28521077572466050961451454007, 3.77704452208817861879520170945, 4.43582013980245260879852653113, 5.17786799960139230407084502020, 6.42844924818217131136636544555, 7.15512410302105794749670597240, 7.65767606149074613295402405098, 9.100249410076355766113954855625, 9.691774144621033406169133897578