| L(s) = 1 | + 83·3-s + 256·4-s − 1.52e3·5-s + 2.18e3·9-s + 2.12e4·12-s − 1.26e5·15-s + 1.63e4·16-s − 3.89e5·20-s + 1.20e6·25-s − 2.72e4·27-s + 3.90e4·31-s + 5.59e5·36-s + 1.12e6·37-s − 3.32e6·45-s − 3.94e6·47-s + 1.35e6·48-s − 1.64e6·49-s + 4.49e5·59-s − 3.23e7·60-s − 4.19e6·64-s + 6.84e5·67-s + 1.00e8·75-s − 2.49e7·80-s − 2.25e6·81-s + 3.24e6·93-s − 1.51e7·97-s + 3.09e8·100-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 2·4-s − 5.44·5-s + 9-s + 3.54·12-s − 9.65·15-s + 16-s − 10.8·20-s + 15.4·25-s − 0.266·27-s + 0.235·31-s + 2·36-s + 3.65·37-s − 5.44·45-s − 5.54·47-s + 1.77·48-s − 2·49-s + 0.285·59-s − 19.3·60-s − 2·64-s + 0.278·67-s + 27.4·75-s − 5.44·80-s − 0.472·81-s + 0.417·93-s − 1.68·97-s + 30.9·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2702659494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2702659494\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - 83 T + 4702 T^{2} - 83 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p^{7} T^{2} + p^{14} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 507 T + p^{7} T^{2} )^{2}( 1 + 507 T + 178924 T^{2} + 507 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 91467 T + 4961386642 T^{2} - 91467 p^{7} T^{3} + p^{14} T^{4} )( 1 + 91467 T + 4961386642 T^{2} + 91467 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 39065 T + p^{7} T^{2} )^{2}( 1 + 39065 T - 25986539886 T^{2} + 39065 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 - 562471 T + 221441748708 T^{2} - 562471 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + 1314924 T + p^{7} T^{2} )^{2}( 1 + 1314924 T + 1222402005313 T^{2} + 1314924 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 363378 T - 1042667568953 T^{2} - 363378 p^{7} T^{3} + p^{14} T^{4} )( 1 + 363378 T - 1042667568953 T^{2} + 363378 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 149955 T + p^{7} T^{2} )^{2}( 1 - 149955 T - 2466164982794 T^{2} - 149955 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 684671 T + p^{7} T^{2} )^{2}( 1 + 684671 T - 5591937227082 T^{2} + 684671 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 71 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 5729217 T + 23728807274698 T^{2} - 5729217 p^{7} T^{3} + p^{14} T^{4} )( 1 + 5729217 T + 23728807274698 T^{2} + 5729217 p^{7} T^{3} + p^{14} T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - p^{7} T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4424121 T + p^{7} T^{2} )^{2}( 1 + 4424121 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 15182479 T + p^{7} T^{2} )^{2}( 1 - 15182479 T + 149709384107328 T^{2} - 15182479 p^{7} T^{3} + p^{14} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.491557460023114514384197763604, −8.340837182526946087717899742537, −8.061650193553308111016984451064, −7.909543503631757894860866297033, −7.67517163172878928487647753876, −7.58609437003464147924333134958, −7.29783354890819329132739804654, −6.79802803983913365004403132573, −6.67461943116270717402862693054, −6.38798144166599954767872671219, −6.00914807817590642921511039673, −4.90769543732250590345112404829, −4.74582911926741768624614542200, −4.59860529058900690391328208241, −4.11347437259348876067503594928, −3.65901587142534059125866227833, −3.54559384820834530739023727526, −3.37985224738418430023288243095, −2.82960075845216663350746326969, −2.71099072513326778122068276502, −2.40384160205204698255364584654, −1.48513756637222296595282318869, −1.26535854721751086734378793490, −0.44737966436465365339191664174, −0.10350946529485303449109421113,
0.10350946529485303449109421113, 0.44737966436465365339191664174, 1.26535854721751086734378793490, 1.48513756637222296595282318869, 2.40384160205204698255364584654, 2.71099072513326778122068276502, 2.82960075845216663350746326969, 3.37985224738418430023288243095, 3.54559384820834530739023727526, 3.65901587142534059125866227833, 4.11347437259348876067503594928, 4.59860529058900690391328208241, 4.74582911926741768624614542200, 4.90769543732250590345112404829, 6.00914807817590642921511039673, 6.38798144166599954767872671219, 6.67461943116270717402862693054, 6.79802803983913365004403132573, 7.29783354890819329132739804654, 7.58609437003464147924333134958, 7.67517163172878928487647753876, 7.909543503631757894860866297033, 8.061650193553308111016984451064, 8.340837182526946087717899742537, 8.491557460023114514384197763604
