| L(s) = 1 | − 8.18·3-s − 16.4·5-s + 7·7-s + 40.0·9-s − 22.4·11-s + 75.2·13-s + 134.·15-s + 53.8·17-s + 96.0·19-s − 57.3·21-s − 174.·23-s + 145.·25-s − 106.·27-s − 139.·29-s − 125.·31-s + 184.·33-s − 115.·35-s + 343.·37-s − 615.·39-s + 445.·41-s − 94.1·43-s − 658.·45-s − 240.·47-s + 49·49-s − 440.·51-s − 419.·53-s + 369.·55-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 1.47·5-s + 0.377·7-s + 1.48·9-s − 0.616·11-s + 1.60·13-s + 2.31·15-s + 0.768·17-s + 1.15·19-s − 0.595·21-s − 1.58·23-s + 1.16·25-s − 0.759·27-s − 0.891·29-s − 0.724·31-s + 0.970·33-s − 0.556·35-s + 1.52·37-s − 2.52·39-s + 1.69·41-s − 0.333·43-s − 2.18·45-s − 0.746·47-s + 0.142·49-s − 1.21·51-s − 1.08·53-s + 0.906·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 + 8.18T + 27T^{2} \) |
| 5 | \( 1 + 16.4T + 125T^{2} \) |
| 11 | \( 1 + 22.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 75.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 96.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 343.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 445.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 94.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 419.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 520.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 540.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 296.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 92.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 122.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.37e3T + 9.12e5T^{2} \) |
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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
−10.78046695165924914403509523129, −9.485025090530543394001249655224, −7.945143594208710074343960729591, −7.66567090523200400083056520998, −6.24483888810221417014420053280, −5.53399109206577260921515647671, −4.42464042633622361616756679310, −3.51524123339726060976881057461, −1.16467260961176665508025566009, 0,
1.16467260961176665508025566009, 3.51524123339726060976881057461, 4.42464042633622361616756679310, 5.53399109206577260921515647671, 6.24483888810221417014420053280, 7.66567090523200400083056520998, 7.945143594208710074343960729591, 9.485025090530543394001249655224, 10.78046695165924914403509523129
