| L(s) = 1 | + 3·3-s + 11.4·7-s + 9·9-s − 16.0·11-s + 73.9·13-s − 103.·17-s − 27.9·19-s + 34.3·21-s − 112.·23-s + 27·27-s − 241.·29-s + 303.·31-s − 48.0·33-s − 177.·37-s + 221.·39-s + 4.91·41-s − 171.·43-s − 373.·47-s − 211.·49-s − 311.·51-s + 45.4·53-s − 83.8·57-s − 711.·59-s + 567.·61-s + 103.·63-s − 390.·67-s − 338.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.618·7-s + 0.333·9-s − 0.438·11-s + 1.57·13-s − 1.48·17-s − 0.337·19-s + 0.357·21-s − 1.02·23-s + 0.192·27-s − 1.54·29-s + 1.76·31-s − 0.253·33-s − 0.789·37-s + 0.910·39-s + 0.0187·41-s − 0.607·43-s − 1.15·47-s − 0.617·49-s − 0.855·51-s + 0.117·53-s − 0.194·57-s − 1.56·59-s + 1.19·61-s + 0.206·63-s − 0.712·67-s − 0.590·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 11.4T + 343T^{2} \) |
| 11 | \( 1 + 16.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 73.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 4.91T + 6.89e4T^{2} \) |
| 43 | \( 1 + 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 45.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 711.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 567.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 390.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 687.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 431.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 720.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 633.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 219.T + 9.12e5T^{2} \) |
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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
−8.377178375290805644172035045924, −7.66966497421114925557130701121, −6.62383387498186539208574223553, −6.02880192843609899162940901003, −4.90413544587068315971811583780, −4.15766420687034564263200125908, −3.33542542118101391845071967878, −2.19288916306872253671018507588, −1.46682280183164767247063406104, 0,
1.46682280183164767247063406104, 2.19288916306872253671018507588, 3.33542542118101391845071967878, 4.15766420687034564263200125908, 4.90413544587068315971811583780, 6.02880192843609899162940901003, 6.62383387498186539208574223553, 7.66966497421114925557130701121, 8.377178375290805644172035045924
