| L(s) = 1 | + 2·3-s − 5-s − 7-s + 9-s + 2·11-s − 2·15-s + 17-s − 4·19-s − 2·21-s + 25-s − 4·27-s + 5·29-s + 7·31-s + 4·33-s + 35-s − 37-s + 5·41-s + 4·43-s − 45-s + 4·47-s − 6·49-s + 2·51-s − 53-s − 2·55-s − 8·57-s + 9·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.516·15-s + 0.242·17-s − 0.917·19-s − 0.436·21-s + 1/5·25-s − 0.769·27-s + 0.928·29-s + 1.25·31-s + 0.696·33-s + 0.169·35-s − 0.164·37-s + 0.780·41-s + 0.609·43-s − 0.149·45-s + 0.583·47-s − 6/7·49-s + 0.280·51-s − 0.137·53-s − 0.269·55-s − 1.05·57-s + 1.17·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.289748923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.289748923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.37188415120003, −12.77279095306013, −12.26186653808886, −11.96809053596702, −11.33521767594507, −10.81373383043587, −10.32742230710027, −9.770527254716075, −9.287959091744343, −8.912674217649472, −8.414844139570830, −7.965050982021402, −7.653050738419767, −6.874001484565278, −6.483748250675451, −6.030340058921024, −5.274136532802731, −4.583738397161156, −4.099271593985788, −3.674743833499928, −3.023873047545739, −2.625611185403884, −2.036857733041684, −1.221471424457389, −0.5037253246795998,
0.5037253246795998, 1.221471424457389, 2.036857733041684, 2.625611185403884, 3.023873047545739, 3.674743833499928, 4.099271593985788, 4.583738397161156, 5.274136532802731, 6.030340058921024, 6.483748250675451, 6.874001484565278, 7.653050738419767, 7.965050982021402, 8.414844139570830, 8.912674217649472, 9.287959091744343, 9.770527254716075, 10.32742230710027, 10.81373383043587, 11.33521767594507, 11.96809053596702, 12.26186653808886, 12.77279095306013, 13.37188415120003
