| L(s) = 1 | + 2-s + 2·4-s + 6·5-s + 5·8-s + 6·10-s + 2·11-s − 2·13-s + 5·16-s + 6·17-s + 4·19-s + 12·20-s + 2·22-s + 2·23-s + 17·25-s − 2·26-s − 2·29-s + 2·31-s + 10·32-s + 6·34-s + 2·37-s + 4·38-s + 30·40-s − 4·41-s − 6·43-s + 4·44-s + 2·46-s + 12·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 2.68·5-s + 1.76·8-s + 1.89·10-s + 0.603·11-s − 0.554·13-s + 5/4·16-s + 1.45·17-s + 0.917·19-s + 2.68·20-s + 0.426·22-s + 0.417·23-s + 17/5·25-s − 0.392·26-s − 0.371·29-s + 0.359·31-s + 1.76·32-s + 1.02·34-s + 0.328·37-s + 0.648·38-s + 4.74·40-s − 0.624·41-s − 0.914·43-s + 0.603·44-s + 0.294·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.32015746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.32015746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 110 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 194 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.478249682641496975093706281935, −7.919414712475635389197239872024, −7.54744854258677867373333520729, −7.16633957255057816150726796036, −7.12614621071580458266347643477, −6.43048483615193478246826020789, −6.20690863194529074716862246155, −5.90638330454593477169100488138, −5.45404215258408735940873203538, −5.37073059993786927539341671792, −4.88678931179284937809491967345, −4.52694098368573618655575315692, −3.97484093769016949516940024822, −3.47235489106323898093215025445, −2.91680196088909721219433053948, −2.61349040716235079611381681774, −2.19806791217929370613213391320, −1.56736055160354139438530074819, −1.48015844081657071450979856897, −0.964284910030881116347584445651,
0.964284910030881116347584445651, 1.48015844081657071450979856897, 1.56736055160354139438530074819, 2.19806791217929370613213391320, 2.61349040716235079611381681774, 2.91680196088909721219433053948, 3.47235489106323898093215025445, 3.97484093769016949516940024822, 4.52694098368573618655575315692, 4.88678931179284937809491967345, 5.37073059993786927539341671792, 5.45404215258408735940873203538, 5.90638330454593477169100488138, 6.20690863194529074716862246155, 6.43048483615193478246826020789, 7.12614621071580458266347643477, 7.16633957255057816150726796036, 7.54744854258677867373333520729, 7.919414712475635389197239872024, 8.478249682641496975093706281935
