L(s) = 1 | + (−0.368 − 0.425i)3-s + (0.142 + 0.989i)5-s + (0.755 − 1.65i)7-s + (0.0971 − 0.675i)9-s + (0.368 − 0.425i)15-s + (−0.983 + 0.288i)21-s + (−0.281 − 0.959i)23-s + (−0.959 + 0.281i)25-s + (−0.797 + 0.512i)27-s + (−0.239 − 0.153i)29-s + (1.74 + 0.512i)35-s + (0.118 + 0.822i)41-s + (−0.708 − 0.817i)43-s + 0.682·45-s + 1.97·47-s + ⋯ |
L(s) = 1 | + (−0.368 − 0.425i)3-s + (0.142 + 0.989i)5-s + (0.755 − 1.65i)7-s + (0.0971 − 0.675i)9-s + (0.368 − 0.425i)15-s + (−0.983 + 0.288i)21-s + (−0.281 − 0.959i)23-s + (−0.959 + 0.281i)25-s + (−0.797 + 0.512i)27-s + (−0.239 − 0.153i)29-s + (1.74 + 0.512i)35-s + (0.118 + 0.822i)41-s + (−0.708 − 0.817i)43-s + 0.682·45-s + 1.97·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.077170880\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077170880\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (0.281 + 0.959i)T \) |
good | 3 | \( 1 + (0.368 + 0.425i)T + (-0.142 + 0.989i)T^{2} \) |
| 7 | \( 1 + (-0.755 + 1.65i)T + (-0.654 - 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 17 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 19 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 31 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 37 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 41 | \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 43 | \( 1 + (0.708 + 0.817i)T + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 - 1.97T + T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \) |
| 67 | \( 1 + (-1.45 + 0.425i)T + (0.841 - 0.540i)T^{2} \) |
| 71 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 79 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (0.215 - 1.49i)T + (-0.959 - 0.281i)T^{2} \) |
| 89 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (0.959 - 0.281i)T^{2} \) |
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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
−9.493414204515425724296789897965, −8.258720810831851249103239803839, −7.55248621047181104399830534478, −6.85607473286559386165761539407, −6.43656872978160377755077582656, −5.31021965817142309203060138234, −4.15737103240940645977659628334, −3.57034500134532114598988148888, −2.18184172316727566515777213809, −0.897360344945198490080332165477,
1.64483266637209398094707243886, 2.48486253040365526479663261461, 4.01938730169843131660101588151, 4.93291010659715331857775622987, 5.46387300605167289060296973674, 5.94644023680898749262205412073, 7.42709115399040180457683998106, 8.215816186543718893773103577467, 8.814290415004307482558018255623, 9.450668173526150219153090530597