| L(s) = 1 | + 16·3-s + 94·9-s − 244·11-s − 256·16-s − 96·17-s − 1.63e3·19-s + 752·27-s − 3.90e3·33-s + 4.41e3·41-s − 7.00e3·43-s − 4.09e3·48-s − 1.53e3·51-s − 2.61e4·57-s + 476·59-s + 2.08e4·67-s − 6.81e3·73-s + 8.77e3·81-s + 2.23e4·83-s − 2.59e3·97-s − 2.29e4·99-s + 4.74e4·107-s − 2.82e4·113-s + 2.60e3·121-s + 7.06e4·123-s + 127-s − 1.12e5·129-s + 131-s + ⋯ |
| L(s) = 1 | + 16/9·3-s + 1.16·9-s − 2.01·11-s − 16-s − 0.332·17-s − 4.52·19-s + 1.03·27-s − 3.58·33-s + 2.62·41-s − 3.78·43-s − 1.77·48-s − 0.590·51-s − 8.03·57-s + 0.136·59-s + 4.64·67-s − 1.27·73-s + 1.33·81-s + 3.24·83-s − 0.275·97-s − 2.34·99-s + 4.14·107-s − 2.21·113-s + 0.177·121-s + 4.67·123-s + 6.20e−5·127-s − 6.73·129-s + 5.82e−5·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.01156180956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01156180956\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 96 T + 4608 T^{2} + 96 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p^{4} T^{3} + p^{8} T^{4} )( 1 + 34 T^{2} + p^{8} T^{4} ) \) |
| 5 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 46 T + p^{4} T^{2} )^{2}( 1 + 336 T + 56448 T^{2} + 336 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 816 T + 332928 T^{2} + 816 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4416 T + 9750528 T^{2} - 4416 p^{4} T^{3} + p^{8} T^{4} )( 1 - 4099006 T^{2} + p^{8} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 3502 T + p^{4} T^{2} )^{2}( 1 + 5426402 T^{2} + p^{8} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{8} T^{4} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 238 T + p^{4} T^{2} )^{2}( 1 - 24178078 T^{2} + p^{8} T^{4} ) \) |
| 61 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 10416 T + 54246528 T^{2} - 10416 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 33567554 T^{2} + p^{8} T^{4} )( 1 + 6816 T + 23228928 T^{2} + 6816 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 79 | $C_4\times C_2$ | \( 1 + p^{16} T^{8} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 - 11186 T + p^{4} T^{2} )^{2}( 1 + 30209954 T^{2} + p^{8} T^{4} ) \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 21024 T + 221004288 T^{2} - 21024 p^{4} T^{3} + p^{8} T^{4} )( 1 + 21024 T + 221004288 T^{2} + 21024 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 9982 T + p^{4} T^{2} )^{2}( 1 + 22560 T + 254476800 T^{2} + 22560 p^{4} T^{3} + p^{8} T^{4} ) \) |
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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
−8.726356236287449108226666999592, −8.721130843283387493502342218786, −8.709107446127771066500381049814, −8.151468312108018823343697971170, −7.956573850981954673728610883713, −7.71307078199527084498729981392, −7.55004181143132265350647743499, −6.74746276707172062961653934206, −6.72082962275500266197296412091, −6.41461904857866493447882067998, −6.35058612567106491422197749205, −5.69350586652579772597411949577, −5.24042056263040031097194692446, −4.85110442440502252369141120105, −4.77997722042100191802934198729, −4.16957574741864819376456668670, −4.05652277325018588653247847517, −3.54242171728002663073070264138, −3.10920353526284652753140246752, −2.63822846191891667454581804948, −2.29548738760257873112453390366, −2.09654199614116920748628368697, −2.01427207787120166261608903495, −0.805504157924571871883003792842, −0.01861432465518819192202680513,
0.01861432465518819192202680513, 0.805504157924571871883003792842, 2.01427207787120166261608903495, 2.09654199614116920748628368697, 2.29548738760257873112453390366, 2.63822846191891667454581804948, 3.10920353526284652753140246752, 3.54242171728002663073070264138, 4.05652277325018588653247847517, 4.16957574741864819376456668670, 4.77997722042100191802934198729, 4.85110442440502252369141120105, 5.24042056263040031097194692446, 5.69350586652579772597411949577, 6.35058612567106491422197749205, 6.41461904857866493447882067998, 6.72082962275500266197296412091, 6.74746276707172062961653934206, 7.55004181143132265350647743499, 7.71307078199527084498729981392, 7.956573850981954673728610883713, 8.151468312108018823343697971170, 8.709107446127771066500381049814, 8.721130843283387493502342218786, 8.726356236287449108226666999592
