| L(s) = 1 | + 17·3-s + 50·5-s + 98·7-s + 81·9-s + 73·11-s − 23·13-s + 850·15-s − 263·17-s + 1.66e3·21-s + 1.87e3·25-s − 782·27-s + 1.15e3·29-s + 1.24e3·33-s + 4.90e3·35-s − 391·39-s + 4.05e3·45-s + 3.45e3·47-s + 7.20e3·49-s − 4.47e3·51-s + 3.65e3·55-s + 7.93e3·63-s − 1.15e3·65-s − 2.01e4·71-s − 1.90e4·73-s + 3.18e4·75-s + 7.15e3·77-s − 1.21e4·79-s + ⋯ |
| L(s) = 1 | + 17/9·3-s + 2·5-s + 2·7-s + 9-s + 0.603·11-s − 0.136·13-s + 34/9·15-s − 0.910·17-s + 34/9·21-s + 3·25-s − 1.07·27-s + 1.37·29-s + 1.13·33-s + 4·35-s − 0.257·39-s + 2·45-s + 1.56·47-s + 3·49-s − 1.71·51-s + 1.20·55-s + 2·63-s − 0.272·65-s − 3.99·71-s − 3.56·73-s + 17/3·75-s + 1.20·77-s − 1.94·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(9.704510406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.704510406\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 17 T + 208 T^{2} - 17 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 73 T - 9312 T^{2} - 73 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T - 28032 T^{2} + 23 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 263 T - 14352 T^{2} + 263 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1153 T + 622128 T^{2} - 1153 p^{4} T^{3} + p^{8} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3457 T + 7071168 T^{2} - 3457 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 10078 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9502 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 12167 T + 109085808 T^{2} + 12167 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6382 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 3383 T - 77084592 T^{2} + 3383 p^{4} T^{3} + p^{8} T^{4} \) |
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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
−13.18092728457032057428286539338, −12.22842267548751434000563424221, −11.67382590210507344805627960448, −11.15568400779382472755262446904, −10.36999457922126728949910792594, −10.21156626829884299583795965990, −9.327494329135720947189951110893, −8.902162700244479647023181287920, −8.588576445316415483550990919874, −8.402345718349715339716014147863, −7.36555687806040724450722813696, −7.06805481455513881042124853851, −5.80606376936883882318647082681, −5.79008890306018768315991152409, −4.60570527350632076768633118231, −4.32895683096746881891987256832, −2.77836882375865090235027515920, −2.69584756017582745928478532805, −1.73523048862302992957979228351, −1.39369564600828526654037337111,
1.39369564600828526654037337111, 1.73523048862302992957979228351, 2.69584756017582745928478532805, 2.77836882375865090235027515920, 4.32895683096746881891987256832, 4.60570527350632076768633118231, 5.79008890306018768315991152409, 5.80606376936883882318647082681, 7.06805481455513881042124853851, 7.36555687806040724450722813696, 8.402345718349715339716014147863, 8.588576445316415483550990919874, 8.902162700244479647023181287920, 9.327494329135720947189951110893, 10.21156626829884299583795965990, 10.36999457922126728949910792594, 11.15568400779382472755262446904, 11.67382590210507344805627960448, 12.22842267548751434000563424221, 13.18092728457032057428286539338
