| L(s) = 1 | + 3.25·3-s − 3.04·5-s − 7-s + 7.60·9-s + 5.22·11-s + 3.38·13-s − 9.92·15-s − 2.63·17-s + 3.39·19-s − 3.25·21-s − 23-s + 4.28·25-s + 15.0·27-s − 4.00·29-s + 6.97·31-s + 17.0·33-s + 3.04·35-s − 5.11·37-s + 11.0·39-s − 5.89·41-s − 7.90·43-s − 23.1·45-s + 9.54·47-s + 49-s − 8.59·51-s − 11.8·53-s − 15.9·55-s + ⋯ |
| L(s) = 1 | + 1.88·3-s − 1.36·5-s − 0.377·7-s + 2.53·9-s + 1.57·11-s + 0.938·13-s − 2.56·15-s − 0.640·17-s + 0.778·19-s − 0.710·21-s − 0.208·23-s + 0.857·25-s + 2.88·27-s − 0.743·29-s + 1.25·31-s + 2.96·33-s + 0.515·35-s − 0.841·37-s + 1.76·39-s − 0.920·41-s − 1.20·43-s − 3.45·45-s + 1.39·47-s + 0.142·49-s − 1.20·51-s − 1.62·53-s − 2.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.422419587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.422419587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 11 | \( 1 - 5.22T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 29 | \( 1 + 4.00T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + 5.89T + 41T^{2} \) |
| 43 | \( 1 + 7.90T + 43T^{2} \) |
| 47 | \( 1 - 9.54T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 1.51T + 59T^{2} \) |
| 61 | \( 1 - 1.13T + 61T^{2} \) |
| 67 | \( 1 + 8.80T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 0.693T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 3.23T + 97T^{2} \) |
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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
−10.37186330165409692017724782903, −9.313417959041747244222682775558, −8.799254398550034938157138665578, −8.101878512190974591523459511841, −7.26827124232272544913110102689, −6.49692389149189810873318672029, −4.41891264819040350668245904011, −3.73422265729056473082784377557, −3.14433545721231089990199690398, −1.50492395757221088481247681802,
1.50492395757221088481247681802, 3.14433545721231089990199690398, 3.73422265729056473082784377557, 4.41891264819040350668245904011, 6.49692389149189810873318672029, 7.26827124232272544913110102689, 8.101878512190974591523459511841, 8.799254398550034938157138665578, 9.313417959041747244222682775558, 10.37186330165409692017724782903
