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
−9.450668173526150219153090530597, −8.814290415004307482558018255623, −8.215816186543718893773103577467, −7.42709115399040180457683998106, −5.94644023680898749262205412073, −5.46387300605167289060296973674, −4.93291010659715331857775622987, −4.01938730169843131660101588151, −2.48486253040365526479663261461, −1.64483266637209398094707243886,
0.897360344945198490080332165477, 2.18184172316727566515777213809, 3.57034500134532114598988148888, 4.15737103240940645977659628334, 5.31021965817142309203060138234, 6.43656872978160377755077582656, 6.85607473286559386165761539407, 7.55248621047181104399830534478, 8.258720810831851249103239803839, 9.493414204515425724296789897965