| L(s) = 1 | + 12·11-s + 4·13-s − 24·23-s − 2·25-s + 8·37-s + 24·47-s + 4·49-s + 12·59-s − 4·61-s + 12·71-s + 4·73-s − 32·97-s − 48·107-s + 4·109-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 48·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + ⋯ |
| L(s) = 1 | + 3.61·11-s + 1.10·13-s − 5.00·23-s − 2/5·25-s + 1.31·37-s + 3.50·47-s + 4/7·49-s + 1.56·59-s − 0.512·61-s + 1.42·71-s + 0.468·73-s − 3.24·97-s − 4.64·107-s + 0.383·109-s + 4.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.196581796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.196581796\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 579 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 12 T + 79 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 1194 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 82 T^{2} + 3171 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4506 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 1002 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 86 T^{2} + 3579 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 202 T^{2} + 20955 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11994 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−6.58419375687555026679701157218, −6.08848902854573756044185132745, −6.06680025212631090811517403248, −5.82657391349391843484439766218, −5.69630346055068226369794198039, −5.67740047924299222287591487497, −5.36834288999675612483169787054, −4.90332606115584730836988250686, −4.59701210179266857549394732317, −4.20655347708816041475239438060, −4.17986634769664571409090512376, −4.00356594034588411006981372264, −3.98952132474515535591517576670, −3.79616696703649373667506068245, −3.58413243263072182514323428625, −3.33149808263759768275105894124, −2.77039976744070424873159568624, −2.38417746702589173654462851348, −2.32550684349787033502632185703, −2.19874616451576287335400237850, −1.48999441325242354282972106993, −1.48164652490249028041693153842, −1.30780022258140895828684577990, −0.813451255177655805913773751107, −0.35413234837232041830278530207,
0.35413234837232041830278530207, 0.813451255177655805913773751107, 1.30780022258140895828684577990, 1.48164652490249028041693153842, 1.48999441325242354282972106993, 2.19874616451576287335400237850, 2.32550684349787033502632185703, 2.38417746702589173654462851348, 2.77039976744070424873159568624, 3.33149808263759768275105894124, 3.58413243263072182514323428625, 3.79616696703649373667506068245, 3.98952132474515535591517576670, 4.00356594034588411006981372264, 4.17986634769664571409090512376, 4.20655347708816041475239438060, 4.59701210179266857549394732317, 4.90332606115584730836988250686, 5.36834288999675612483169787054, 5.67740047924299222287591487497, 5.69630346055068226369794198039, 5.82657391349391843484439766218, 6.06680025212631090811517403248, 6.08848902854573756044185132745, 6.58419375687555026679701157218
