| L(s) = 1 | + 87·2-s + 1.63e4·4-s + 3.61e6·8-s − 4.78e6·9-s − 3.64e7·11-s + 3.14e8·16-s − 4.16e8·18-s − 3.17e9·22-s + 2.18e9·23-s − 6.10e9·25-s + 5.96e10·29-s + 5.92e10·32-s − 7.83e10·36-s + 1.12e11·37-s + 9.69e11·43-s − 5.96e11·44-s + 1.90e11·46-s − 5.31e11·50-s − 9.07e11·53-s + 5.18e12·58-s + 8.68e12·64-s − 1.15e13·67-s − 8.67e12·71-s − 1.73e13·72-s + 9.77e12·74-s + 3.71e13·79-s + 8.43e13·86-s + ⋯ |
| L(s) = 1 | + 0.679·2-s + 4-s + 1.72·8-s − 9-s − 1.86·11-s + 1.17·16-s − 0.679·18-s − 1.27·22-s + 0.642·23-s − 25-s + 3.45·29-s + 1.72·32-s − 36-s + 1.18·37-s + 3.56·43-s − 1.86·44-s + 0.436·46-s − 0.679·50-s − 0.772·53-s + 2.35·58-s + 1.97·64-s − 1.90·67-s − 0.954·71-s − 1.72·72-s + 0.804·74-s + 1.93·79-s + 2.42·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+7)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(5.473011172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.473011172\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 87 T - 8815 T^{2} - 87 p^{14} T^{3} + p^{28} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 36437514 T + 947942592916955 T^{2} + 36437514 p^{14} T^{3} + p^{28} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2188914318 T - 6801490432993344685 T^{2} - 2188914318 p^{14} T^{3} + p^{28} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 29824366266 T + p^{14} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 112367216342 T + \)\(36\!\cdots\!75\)\( T^{2} - 112367216342 p^{14} T^{3} + p^{28} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 484972531402 T + p^{14} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 907194972426 T - \)\(55\!\cdots\!93\)\( T^{2} + 907194972426 p^{14} T^{3} + p^{28} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 11528240589818 T + \)\(96\!\cdots\!95\)\( T^{2} + 11528240589818 p^{14} T^{3} + p^{28} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4338861915246 T + p^{14} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 37193960502814 T + \)\(10\!\cdots\!15\)\( T^{2} - 37193960502814 p^{14} T^{3} + p^{28} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
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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
−13.00924184464706143174777914087, −12.30182852530922819703260760814, −11.81780178964798492708915927634, −11.10151124786747150556653413286, −10.53380732252978826922274041112, −10.46143642602162148389493055176, −9.462454758675497209776033463287, −8.528785318473427985004614505465, −7.75319713827998740969943256065, −7.73641796467974542332413836206, −6.74207209244057095229551477893, −6.01500076689465439096505285114, −5.54919151433852342396893344636, −4.63343895730718216703820478134, −4.43999616964407143782063933451, −3.12512105726019770350082469791, −2.60943288873292470019037501051, −2.33149123698710356932101286056, −1.16252665383595614249840974699, −0.54897576081924540040719238838,
0.54897576081924540040719238838, 1.16252665383595614249840974699, 2.33149123698710356932101286056, 2.60943288873292470019037501051, 3.12512105726019770350082469791, 4.43999616964407143782063933451, 4.63343895730718216703820478134, 5.54919151433852342396893344636, 6.01500076689465439096505285114, 6.74207209244057095229551477893, 7.73641796467974542332413836206, 7.75319713827998740969943256065, 8.528785318473427985004614505465, 9.462454758675497209776033463287, 10.46143642602162148389493055176, 10.53380732252978826922274041112, 11.10151124786747150556653413286, 11.81780178964798492708915927634, 12.30182852530922819703260760814, 13.00924184464706143174777914087
