| L(s) = 1 | − 9.54·3-s + 15.8·5-s + 64.0·9-s − 21.7·11-s − 21.3·13-s − 150.·15-s − 75.7·17-s − 21.2·19-s + 102.·23-s + 124.·25-s − 353.·27-s + 91.0·29-s + 66.8·31-s + 208.·33-s − 168.·37-s + 203.·39-s − 101.·41-s + 314.·43-s + 1.01e3·45-s + 269.·47-s + 723.·51-s + 308.·53-s − 344.·55-s + 203.·57-s − 808.·59-s − 653.·61-s − 337.·65-s + ⋯ |
| L(s) = 1 | − 1.83·3-s + 1.41·5-s + 2.37·9-s − 0.597·11-s − 0.455·13-s − 2.59·15-s − 1.08·17-s − 0.257·19-s + 0.927·23-s + 0.999·25-s − 2.52·27-s + 0.582·29-s + 0.387·31-s + 1.09·33-s − 0.749·37-s + 0.836·39-s − 0.386·41-s + 1.11·43-s + 3.35·45-s + 0.835·47-s + 1.98·51-s + 0.800·53-s − 0.844·55-s + 0.472·57-s − 1.78·59-s − 1.37·61-s − 0.643·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 9.54T + 27T^{2} \) |
| 5 | \( 1 - 15.8T + 125T^{2} \) |
| 11 | \( 1 + 21.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 75.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 66.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 101.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 314.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 308.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 808.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 653.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 473.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 157.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 270.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 325.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 149.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.86e3T + 9.12e5T^{2} \) |
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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
−9.764735272874706297503887207132, −8.890419688107218583602598573320, −7.34360747742967454861581124188, −6.56569440569713870297853119912, −5.90225765126462179774303280014, −5.16373407623290617315156685419, −4.47278941957814084144023490126, −2.49931006589019477622607094301, −1.31410746435531060407858701246, 0,
1.31410746435531060407858701246, 2.49931006589019477622607094301, 4.47278941957814084144023490126, 5.16373407623290617315156685419, 5.90225765126462179774303280014, 6.56569440569713870297853119912, 7.34360747742967454861581124188, 8.890419688107218583602598573320, 9.764735272874706297503887207132
