| L(s) = 1 | + 27·3-s − 100·5-s − 2.77e3·11-s − 6.58e3·13-s − 2.70e3·15-s − 5.90e3·17-s − 6.64e3·19-s − 1.98e3·23-s + 7.81e4·25-s − 1.96e4·27-s − 4.16e5·29-s + 1.17e5·31-s − 7.48e4·33-s + 3.35e5·37-s − 1.77e5·39-s − 5.30e5·41-s − 1.86e5·43-s + 6.57e5·47-s − 1.59e5·51-s + 6.08e5·53-s + 2.77e5·55-s − 1.79e5·57-s + 5.36e5·59-s + 1.79e6·61-s + 6.58e5·65-s − 2.12e6·67-s − 5.35e4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.357·5-s − 0.628·11-s − 0.831·13-s − 0.206·15-s − 0.291·17-s − 0.222·19-s − 0.0339·23-s + 25-s − 0.192·27-s − 3.16·29-s + 0.710·31-s − 0.362·33-s + 1.08·37-s − 0.480·39-s − 1.20·41-s − 0.357·43-s + 0.923·47-s − 0.168·51-s + 0.561·53-s + 0.224·55-s − 0.128·57-s + 0.339·59-s + 1.01·61-s + 0.297·65-s − 0.862·67-s − 0.0196·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.804682592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.804682592\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 p^{2} T - 109 p^{4} T^{2} + 4 p^{9} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2774 T - 11792095 T^{2} + 2774 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3294 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 5900 T - 375528673 T^{2} + 5900 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6644 T - 849729003 T^{2} + 6644 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 1982 T - 3400897123 T^{2} + 1982 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 208106 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 117792 T - 13637658847 T^{2} - 117792 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 335686 T + 17753213463 T^{2} - 335686 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 265488 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 93292 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 657516 T - 74295830207 T^{2} - 657516 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 608718 T - 804173536313 T^{2} - 608718 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 536120 T - 2201226830419 T^{2} - 536120 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 1797090 T + 86789632079 T^{2} - 1797090 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2123176 T - 1552835278347 T^{2} + 2123176 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1191214 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1056430 T - 9931354174197 T^{2} + 1056430 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 998484 T - 18206938687903 T^{2} + 998484 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3898004 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4622352 T - 22865196883625 T^{2} - 4622352 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 15287710 T + p^{7} T^{2} )^{2} \) |
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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
−9.526310789008828306810841051286, −9.495818987264873884522411299940, −8.785263276893203329811772806620, −8.616739772922722157718291234629, −7.81551376058665930169679895286, −7.79158005181949934800826407394, −7.05102207197775877800884172710, −7.02072238307116102495631711165, −6.06285218555691671597483061057, −5.80497468872726357241131205572, −4.96147929170139513697898724779, −4.95749975630065742813192181952, −4.02602473199963266099005443967, −3.82208079131516615931526356892, −3.02111947435136813503268888911, −2.75950054500881910774087486123, −1.92356497833029215926701013168, −1.85958056382846429834314100631, −0.67074930057982823709831660355, −0.41759672478793209281249335355,
0.41759672478793209281249335355, 0.67074930057982823709831660355, 1.85958056382846429834314100631, 1.92356497833029215926701013168, 2.75950054500881910774087486123, 3.02111947435136813503268888911, 3.82208079131516615931526356892, 4.02602473199963266099005443967, 4.95749975630065742813192181952, 4.96147929170139513697898724779, 5.80497468872726357241131205572, 6.06285218555691671597483061057, 7.02072238307116102495631711165, 7.05102207197775877800884172710, 7.79158005181949934800826407394, 7.81551376058665930169679895286, 8.616739772922722157718291234629, 8.785263276893203329811772806620, 9.495818987264873884522411299940, 9.526310789008828306810841051286
