| L(s) = 1 | + 9·2-s + 64·4-s + 686·7-s + 999·8-s + 1.45e3·9-s − 1.96e3·11-s + 6.17e3·14-s + 8.99e3·16-s + 1.31e4·18-s − 1.76e4·22-s − 2.27e4·23-s + 4.39e4·28-s + 2.12e4·29-s + 6.39e4·32-s + 9.33e4·36-s + 1.01e5·37-s − 1.26e5·43-s − 1.25e5·44-s − 2.04e5·46-s + 3.52e5·49-s − 1.00e5·53-s + 6.85e5·56-s + 1.90e5·58-s + 1.00e6·63-s + 7.35e5·64-s − 5.39e4·67-s + 2.42e5·71-s + ⋯ |
| L(s) = 1 | + 9/8·2-s + 4-s + 2·7-s + 1.95·8-s + 2·9-s − 1.47·11-s + 9/4·14-s + 2.19·16-s + 9/4·18-s − 1.65·22-s − 1.86·23-s + 2·28-s + 0.870·29-s + 1.95·32-s + 2·36-s + 1.99·37-s − 1.59·43-s − 1.47·44-s − 2.10·46-s + 3·49-s − 0.676·53-s + 3.90·56-s + 0.978·58-s + 4·63-s + 2.80·64-s − 0.179·67-s + 0.677·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(12.33809130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.33809130\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 9 T + 17 T^{2} - 9 p^{6} T^{3} + p^{12} T^{4} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 1962 T + 2077883 T^{2} + 1962 p^{6} T^{3} + p^{12} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22734 T + 368798867 T^{2} + 22734 p^{6} T^{3} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 21222 T - 144450037 T^{2} - 21222 p^{6} T^{3} + p^{12} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 101194 T + 7674499227 T^{2} - 101194 p^{6} T^{3} + p^{12} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 126614 T + 9709741947 T^{2} + 126614 p^{6} T^{3} + p^{12} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 50346 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 53926 T - 87550368693 T^{2} + 53926 p^{6} T^{3} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 242478 T - 69304703437 T^{2} - 242478 p^{6} T^{3} + p^{12} T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 929378 T + 620656011363 T^{2} + 929378 p^{6} T^{3} + p^{12} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
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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
−11.87571844644012939985111775597, −11.44168419803604828047581385289, −10.91826039216407037045905024977, −10.31170576690093125838283591217, −10.25402706691347069344975855059, −9.692380393121204521706619485416, −8.446450334340362919209316685461, −7.85589338207442810001290288137, −7.84492015782505743020605100596, −7.32861497145320975896700038367, −6.62652222400842244164080371089, −5.84900786598581218173981214644, −4.98667622151302245815313843726, −4.92740244238642740954427059461, −4.17059008144428239657965861760, −4.06935132221113945759083612116, −2.71728702404284806646519914351, −1.88912714160864088651510401664, −1.64934662380250871317743031300, −0.861365160184202740004068821487,
0.861365160184202740004068821487, 1.64934662380250871317743031300, 1.88912714160864088651510401664, 2.71728702404284806646519914351, 4.06935132221113945759083612116, 4.17059008144428239657965861760, 4.92740244238642740954427059461, 4.98667622151302245815313843726, 5.84900786598581218173981214644, 6.62652222400842244164080371089, 7.32861497145320975896700038367, 7.84492015782505743020605100596, 7.85589338207442810001290288137, 8.446450334340362919209316685461, 9.692380393121204521706619485416, 10.25402706691347069344975855059, 10.31170576690093125838283591217, 10.91826039216407037045905024977, 11.44168419803604828047581385289, 11.87571844644012939985111775597
