| L(s) = 1 | + 81·3-s + 625·5-s − 1.83e3·7-s + 6.56e3·9-s − 4.43e4·11-s + 1.36e5·13-s + 5.06e4·15-s − 2.53e5·17-s − 8.54e4·19-s − 1.49e5·21-s + 9.79e5·23-s + 3.90e5·25-s + 5.31e5·27-s + 2.58e6·29-s − 8.94e6·31-s − 3.59e6·33-s − 1.14e6·35-s + 1.56e7·37-s + 1.10e7·39-s + 2.44e7·41-s − 1.27e7·43-s + 4.10e6·45-s + 6.16e7·47-s − 3.69e7·49-s − 2.05e7·51-s + 5.70e6·53-s − 2.77e7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.289·7-s + 0.333·9-s − 0.914·11-s + 1.32·13-s + 0.258·15-s − 0.736·17-s − 0.150·19-s − 0.167·21-s + 0.729·23-s + 0.200·25-s + 0.192·27-s + 0.679·29-s − 1.74·31-s − 0.527·33-s − 0.129·35-s + 1.36·37-s + 0.765·39-s + 1.35·41-s − 0.569·43-s + 0.149·45-s + 1.84·47-s − 0.916·49-s − 0.425·51-s + 0.0993·53-s − 0.408·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.975927121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.975927121\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 81T \) |
| 5 | \( 1 - 625T \) |
| good | 7 | \( 1 + 1.83e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.43e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.36e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.54e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.79e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.94e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.56e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.44e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.27e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.16e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.70e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.35e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.48e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.43e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.19e9T + 7.60e17T^{2} \) |
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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
−10.55554373611295785407520660318, −9.394677083226720308408694054961, −8.690407304026890060138233588879, −7.65882209329884686082064782076, −6.53025663914131610330986049956, −5.52894401786957868887807603706, −4.22224940307080564205864670562, −3.06350901110700000446908845233, −2.06976882445125676939155175576, −0.77640565276540983474452971112,
0.77640565276540983474452971112, 2.06976882445125676939155175576, 3.06350901110700000446908845233, 4.22224940307080564205864670562, 5.52894401786957868887807603706, 6.53025663914131610330986049956, 7.65882209329884686082064782076, 8.690407304026890060138233588879, 9.394677083226720308408694054961, 10.55554373611295785407520660318
