| L(s) = 1 | + 4·5-s + 8·9-s − 8·11-s − 4·19-s + 2·25-s + 8·29-s − 16·31-s − 8·41-s + 32·45-s + 4·49-s − 32·55-s − 16·59-s + 30·81-s + 8·89-s − 16·95-s − 64·99-s − 32·101-s + 8·109-s + 12·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 8/3·9-s − 2.41·11-s − 0.917·19-s + 2/5·25-s + 1.48·29-s − 2.87·31-s − 1.24·41-s + 4.77·45-s + 4/7·49-s − 4.31·55-s − 2.08·59-s + 10/3·81-s + 0.847·89-s − 1.64·95-s − 6.43·99-s − 3.18·101-s + 0.766·109-s + 1.09·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.387127433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.387127433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 706 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 7330 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11014 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 4066 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25030 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2134 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 216 T^{2} + 23282 T^{4} - 216 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.80283812598874048057728621699, −6.68576595366875872954711753345, −6.45414925598968588460083968723, −5.92079966071911886480834971681, −5.73559224303375853036450137298, −5.65481873053129725091626738544, −5.62415784219016554791719470489, −5.17736046346723719750795422791, −5.15389598807321655221649566183, −4.73297436810243206945379095644, −4.48621100213572687884157417519, −4.36236879414374898923380391109, −4.24250849278742007502858327741, −3.91654338685982175245295465805, −3.40030395237820032372923260713, −3.37469346335069180942106210771, −2.87267544708542629005470305947, −2.82162875562762919341436214958, −2.41739590564268304983929621571, −1.91137064637772366362919879439, −1.80805498302339497783668564638, −1.76237579290334881151119298813, −1.60595494601426299640296757023, −0.75871766723534815344560609660, −0.41770907532909417106655681277,
0.41770907532909417106655681277, 0.75871766723534815344560609660, 1.60595494601426299640296757023, 1.76237579290334881151119298813, 1.80805498302339497783668564638, 1.91137064637772366362919879439, 2.41739590564268304983929621571, 2.82162875562762919341436214958, 2.87267544708542629005470305947, 3.37469346335069180942106210771, 3.40030395237820032372923260713, 3.91654338685982175245295465805, 4.24250849278742007502858327741, 4.36236879414374898923380391109, 4.48621100213572687884157417519, 4.73297436810243206945379095644, 5.15389598807321655221649566183, 5.17736046346723719750795422791, 5.62415784219016554791719470489, 5.65481873053129725091626738544, 5.73559224303375853036450137298, 5.92079966071911886480834971681, 6.45414925598968588460083968723, 6.68576595366875872954711753345, 6.80283812598874048057728621699
