| L(s) = 1 | − 8·2-s + 66·3-s + 400·5-s − 528·6-s + 512·8-s + 2.18e3·9-s − 3.20e3·10-s − 40·11-s − 8.90e3·13-s + 2.64e4·15-s − 4.09e3·16-s − 3.65e4·17-s − 1.74e4·18-s + 4.62e4·19-s + 320·22-s + 1.05e5·23-s + 3.37e4·24-s + 7.81e4·25-s + 7.12e4·26-s + 1.45e5·27-s − 2.52e5·29-s − 2.11e5·30-s + 1.70e5·31-s − 2.64e3·33-s + 2.92e5·34-s − 2.09e4·37-s − 3.69e5·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.41·3-s + 1.43·5-s − 0.997·6-s + 0.353·8-s + 9-s − 1.01·10-s − 0.00906·11-s − 1.12·13-s + 2.01·15-s − 1/4·16-s − 1.80·17-s − 0.707·18-s + 1.54·19-s + 0.00640·22-s + 1.80·23-s + 0.498·24-s + 25-s + 0.794·26-s + 1.42·27-s − 1.92·29-s − 1.42·30-s + 1.03·31-s − 0.0127·33-s + 1.27·34-s − 0.0680·37-s − 1.09·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.065802556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.065802556\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{3} T + p^{6} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 22 p T + 241 p^{2} T^{2} - 22 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 16 p^{2} T + 131 p^{4} T^{2} - 16 p^{9} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 40 T - 19485571 T^{2} + 40 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4452 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 36502 T + 922057331 T^{2} + 36502 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 46222 T + 1242601545 T^{2} - 46222 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 105200 T + 7662214553 T^{2} - 105200 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 126334 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 170964 T + 1716075185 T^{2} - 170964 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 20954 T - 94492807017 T^{2} + 20954 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 318486 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1808 p T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 703716 T - 11406911807 T^{2} + 703716 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1603278 T + 1395789205447 T^{2} + 1603278 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1171894 T - 1115315937583 T^{2} - 1171894 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2068872 T + 1137488516363 T^{2} - 2068872 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 994268 T - 5072142749499 T^{2} - 994268 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 33280 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2971454 T - 2217859644981 T^{2} - 2971454 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2376168 T - 13557734621935 T^{2} - 2376168 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2122358 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6920346 T + 3659853864187 T^{2} + 6920346 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4952710 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.80556412157439152715332776653, −12.71798536492140488208694739217, −11.45754641203427783189494341253, −11.12006046780482725043634803291, −10.37958399660676031826455772036, −9.721095311494831980203993918433, −9.352434211248108905835510625287, −9.249075449276334173847014611480, −8.551417624636166849603117771542, −7.972857135348102998188571626312, −7.15930104537138003496908064394, −6.92886385706456881129938171718, −5.98008452755915705520309190509, −5.06323636839422454342614967735, −4.68039065963113970568763950107, −3.50203599845242979022581034655, −2.65617441204906230801504865997, −2.32575254947735850109986168913, −1.51331174177085880622508868951, −0.65185131998312582174769846655,
0.65185131998312582174769846655, 1.51331174177085880622508868951, 2.32575254947735850109986168913, 2.65617441204906230801504865997, 3.50203599845242979022581034655, 4.68039065963113970568763950107, 5.06323636839422454342614967735, 5.98008452755915705520309190509, 6.92886385706456881129938171718, 7.15930104537138003496908064394, 7.972857135348102998188571626312, 8.551417624636166849603117771542, 9.249075449276334173847014611480, 9.352434211248108905835510625287, 9.721095311494831980203993918433, 10.37958399660676031826455772036, 11.12006046780482725043634803291, 11.45754641203427783189494341253, 12.71798536492140488208694739217, 12.80556412157439152715332776653
