| L(s) = 1 | + 6·4-s − 6·9-s + 13·16-s − 36·36-s + 6·64-s + 9·81-s − 72·121-s + 127-s + 131-s + 137-s + 139-s − 78·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 3·4-s − 2·9-s + 13/4·16-s − 6·36-s + 3/4·64-s + 81-s − 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.5·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.608173282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.608173282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 - 3 T^{2} + 7 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 2 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 36 T^{2} + 553 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 36 T^{2} + 2113 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4370 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 + 204 T^{2} + 17353 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 70 T^{2} + 1179 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 204 T^{2} + 18913 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 13730 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 276 T^{2} + 33313 T^{4} - 276 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.88905168787868174952434321938, −4.64173271410630177290949721004, −4.44691056243219158394704734435, −4.35260265614653042536752890844, −4.24515812705287183770135197555, −4.03409763126126872004764144372, −4.03031172764171224342283785422, −3.70241138066058293985744316849, −3.51062001394689198702561074938, −3.30848896402607797933742085126, −3.24332505869144330039028124369, −2.97229478069689275858734346527, −2.91522817695959817203080778591, −2.83894836680697023088072432065, −2.69961439291851440483029780046, −2.67917700727704387932272091423, −2.19355727497122378676402433236, −2.12727199003770970708159030772, −1.93847432274855684592156995941, −1.76113247808402010973505500028, −1.72473322487527785756129841546, −1.61952475841461065483550852555, −0.901807985891114613638592164840, −0.64783047454391496623114860193, −0.48634814710896855157750741463,
0.48634814710896855157750741463, 0.64783047454391496623114860193, 0.901807985891114613638592164840, 1.61952475841461065483550852555, 1.72473322487527785756129841546, 1.76113247808402010973505500028, 1.93847432274855684592156995941, 2.12727199003770970708159030772, 2.19355727497122378676402433236, 2.67917700727704387932272091423, 2.69961439291851440483029780046, 2.83894836680697023088072432065, 2.91522817695959817203080778591, 2.97229478069689275858734346527, 3.24332505869144330039028124369, 3.30848896402607797933742085126, 3.51062001394689198702561074938, 3.70241138066058293985744316849, 4.03031172764171224342283785422, 4.03409763126126872004764144372, 4.24515812705287183770135197555, 4.35260265614653042536752890844, 4.44691056243219158394704734435, 4.64173271410630177290949721004, 4.88905168787868174952434321938
Plot not available for L-functions of degree greater than 10.
