| L(s) = 1 | + 3.16·3-s + 5-s − 3.96·7-s + 6.99·9-s − 3.65·11-s + 0.463·13-s + 3.16·15-s + 1.06·17-s + 1.84·19-s − 12.5·21-s − 7.33·23-s + 25-s + 12.6·27-s + 8.99·29-s + 5.89·31-s − 11.5·33-s − 3.96·35-s + 8.75·37-s + 1.46·39-s + 4.36·41-s − 0.882·43-s + 6.99·45-s + 8.83·47-s + 8.70·49-s + 3.36·51-s + 5.69·53-s − 3.65·55-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 0.447·5-s − 1.49·7-s + 2.33·9-s − 1.10·11-s + 0.128·13-s + 0.816·15-s + 0.258·17-s + 0.423·19-s − 2.73·21-s − 1.52·23-s + 0.200·25-s + 2.43·27-s + 1.66·29-s + 1.05·31-s − 2.01·33-s − 0.669·35-s + 1.43·37-s + 0.234·39-s + 0.680·41-s − 0.134·43-s + 1.04·45-s + 1.28·47-s + 1.24·49-s + 0.470·51-s + 0.782·53-s − 0.493·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.816276964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.816276964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 - 0.463T + 13T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 - 8.99T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 - 8.75T + 37T^{2} \) |
| 41 | \( 1 - 4.36T + 41T^{2} \) |
| 43 | \( 1 + 0.882T + 43T^{2} \) |
| 47 | \( 1 - 8.83T + 47T^{2} \) |
| 53 | \( 1 - 5.69T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 + 6.45T + 61T^{2} \) |
| 67 | \( 1 + 8.13T + 67T^{2} \) |
| 71 | \( 1 - 1.36T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 4.23T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.78657655485448573048205462197, −7.49134121723652007989684842190, −6.38570105474287447772546391311, −6.06237394896185369646768550294, −4.82213657392098542554182160262, −4.04479477699467947303214572737, −3.24398289996172212934598322103, −2.71423522019830511800303653768, −2.22619266583845741800601801305, −0.871137683853984372333855246505,
0.871137683853984372333855246505, 2.22619266583845741800601801305, 2.71423522019830511800303653768, 3.24398289996172212934598322103, 4.04479477699467947303214572737, 4.82213657392098542554182160262, 6.06237394896185369646768550294, 6.38570105474287447772546391311, 7.49134121723652007989684842190, 7.78657655485448573048205462197
