| 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
| \(L(4)\) |
\(\approx\) |
\(0.1233940028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1233940028\) |
| \(L(\frac{9}{2})\) |
|
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
−12.48346560398588895539880741823, −12.25537126037104317625090542957, −11.48412836126144715154916409472, −11.07354074810820030361304728783, −10.90669482047203838526360689586, −10.35584725453718489116231609358, −9.470007529855321400425155368275, −8.980028474831705921957217057880, −8.361573933128953150813374274363, −7.54232579189291028114852102925, −7.49791670691987972983367403836, −6.55955014817735179165764536308, −5.89392227259618397761829601106, −5.31683840427110438640773225745, −4.60507276980472473929963980129, −3.72737412172700193262392760440, −3.39758568627401511175543601775, −1.73130725810530651235184728536, −1.00808163913224526341540845644, −0.17268254506206528242897929979,
0.17268254506206528242897929979, 1.00808163913224526341540845644, 1.73130725810530651235184728536, 3.39758568627401511175543601775, 3.72737412172700193262392760440, 4.60507276980472473929963980129, 5.31683840427110438640773225745, 5.89392227259618397761829601106, 6.55955014817735179165764536308, 7.49791670691987972983367403836, 7.54232579189291028114852102925, 8.361573933128953150813374274363, 8.980028474831705921957217057880, 9.470007529855321400425155368275, 10.35584725453718489116231609358, 10.90669482047203838526360689586, 11.07354074810820030361304728783, 11.48412836126144715154916409472, 12.25537126037104317625090542957, 12.48346560398588895539880741823
