| L(s) = 1 | − 1.70·3-s + 5-s + 3.70·7-s − 0.0783·9-s + 4.63·11-s − 4.34·13-s − 1.70·15-s + 2.63·17-s + 5.70·19-s − 6.34·21-s + 4.78·23-s + 25-s + 5.26·27-s − 29-s − 10.3·31-s − 7.91·33-s + 3.70·35-s + 6.04·37-s + 7.41·39-s − 9.60·41-s + 5.70·43-s − 0.0783·45-s − 10.3·47-s + 6.75·49-s − 4.49·51-s + 3.07·53-s + 4.63·55-s + ⋯ |
| L(s) = 1 | − 0.986·3-s + 0.447·5-s + 1.40·7-s − 0.0261·9-s + 1.39·11-s − 1.20·13-s − 0.441·15-s + 0.638·17-s + 1.30·19-s − 1.38·21-s + 0.998·23-s + 0.200·25-s + 1.01·27-s − 0.185·29-s − 1.86·31-s − 1.37·33-s + 0.626·35-s + 0.994·37-s + 1.18·39-s − 1.49·41-s + 0.870·43-s − 0.0116·45-s − 1.51·47-s + 0.965·49-s − 0.629·51-s + 0.422·53-s + 0.624·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724923185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724923185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 6.04T + 37T^{2} \) |
| 41 | \( 1 + 9.60T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 3.07T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 8.20T + 67T^{2} \) |
| 71 | \( 1 + 5.75T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 - 1.55T + 83T^{2} \) |
| 89 | \( 1 - 3.57T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−9.147739832905329091708299005959, −8.185935419091983803497888997942, −7.29591875022095861133876568528, −6.72445440295037861296373105634, −5.51571320238036937783075273215, −5.30025853644153223241245211964, −4.45360171894571214656115523475, −3.23567460833641794162329489770, −1.87376336417282290375137325130, −0.957358536807291547678858236720,
0.957358536807291547678858236720, 1.87376336417282290375137325130, 3.23567460833641794162329489770, 4.45360171894571214656115523475, 5.30025853644153223241245211964, 5.51571320238036937783075273215, 6.72445440295037861296373105634, 7.29591875022095861133876568528, 8.185935419091983803497888997942, 9.147739832905329091708299005959
