L(s) = 1 | + 12·4-s − 12·9-s + 74·16-s − 144·36-s − 8·49-s + 300·64-s + 90·81-s − 64·103-s + 36·121-s + 127-s + 131-s + 137-s + 139-s − 888·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 96·196-s + 197-s + ⋯ |
L(s) = 1 | + 6·4-s − 4·9-s + 37/2·16-s − 24·36-s − 8/7·49-s + 75/2·64-s + 10·81-s − 6.30·103-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 74·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 6.85·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.983097162\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.983097162\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 + p T^{2} )^{4} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
good | 2 | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{4} \) |
| 5 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 148 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 168 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−5.07554318920265461928165399321, −5.00479073195260296660613343641, −4.86470556934807182768937891741, −4.38725475015408138264331021898, −4.24838215781594132499593445922, −4.20713247798820858446140678073, −4.01044065715901894027728806960, −3.73418325082588303900545439535, −3.41727692964786015616321500787, −3.38943196111494385344942694132, −3.31793567612243685327676257964, −3.07818691102063799227753410754, −2.91334775514817237785793579861, −2.88939704245747309173082413708, −2.69385462098198913502075904767, −2.67657685307763313167054515563, −2.38675107297204637427583589064, −2.31235355529983806978621431062, −2.12957214607603629877565854009, −1.89472155857754178365895978694, −1.82709154149000353920501128067, −1.56043987756230250197157897733, −1.19583584977527851008393265982, −1.08088981180730255606093189827, −0.29784095610922973919854403482,
0.29784095610922973919854403482, 1.08088981180730255606093189827, 1.19583584977527851008393265982, 1.56043987756230250197157897733, 1.82709154149000353920501128067, 1.89472155857754178365895978694, 2.12957214607603629877565854009, 2.31235355529983806978621431062, 2.38675107297204637427583589064, 2.67657685307763313167054515563, 2.69385462098198913502075904767, 2.88939704245747309173082413708, 2.91334775514817237785793579861, 3.07818691102063799227753410754, 3.31793567612243685327676257964, 3.38943196111494385344942694132, 3.41727692964786015616321500787, 3.73418325082588303900545439535, 4.01044065715901894027728806960, 4.20713247798820858446140678073, 4.24838215781594132499593445922, 4.38725475015408138264331021898, 4.86470556934807182768937891741, 5.00479073195260296660613343641, 5.07554318920265461928165399321
Plot not available for L-functions of degree greater than 10.