| L(s) = 1 | + 0.0875·3-s + 5·5-s + 28.3·7-s − 26.9·9-s + 32.7·11-s + 18.5·13-s + 0.437·15-s + 6.98·17-s + 147.·19-s + 2.48·21-s + 23·23-s + 25·25-s − 4.72·27-s + 287.·29-s − 269.·31-s + 2.87·33-s + 141.·35-s − 84.0·37-s + 1.62·39-s − 126.·41-s − 229.·43-s − 134.·45-s + 110.·47-s + 463.·49-s + 0.611·51-s + 358.·53-s + 163.·55-s + ⋯ |
| L(s) = 1 | + 0.0168·3-s + 0.447·5-s + 1.53·7-s − 0.999·9-s + 0.898·11-s + 0.396·13-s + 0.00753·15-s + 0.0996·17-s + 1.78·19-s + 0.0258·21-s + 0.208·23-s + 0.200·25-s − 0.0337·27-s + 1.84·29-s − 1.56·31-s + 0.0151·33-s + 0.685·35-s − 0.373·37-s + 0.00667·39-s − 0.483·41-s − 0.814·43-s − 0.447·45-s + 0.341·47-s + 1.35·49-s + 0.00167·51-s + 0.928·53-s + 0.401·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.388769951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.388769951\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 0.0875T + 27T^{2} \) |
| 7 | \( 1 - 28.3T + 343T^{2} \) |
| 11 | \( 1 - 32.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.98T + 4.91e3T^{2} \) |
| 19 | \( 1 - 147.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 287.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 269.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 84.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 229.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 358.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 400.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 721.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 590.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 976.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 152.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 817.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−8.774538654005608740674162155144, −8.264363305914355545603443796225, −7.37370097283884915147669961626, −6.49062454923916741911412283791, −5.42977857877429847580333146558, −5.09512035713081351194950807412, −3.88662267161758909310538843798, −2.89360911345742656300234172638, −1.72060290146748587118071598600, −0.939111987001259654128718914486,
0.939111987001259654128718914486, 1.72060290146748587118071598600, 2.89360911345742656300234172638, 3.88662267161758909310538843798, 5.09512035713081351194950807412, 5.42977857877429847580333146558, 6.49062454923916741911412283791, 7.37370097283884915147669961626, 8.264363305914355545603443796225, 8.774538654005608740674162155144
