| L(s) = 1 | − 2·4-s + 5·9-s − 4·11-s + 3·16-s − 12·19-s − 4·29-s − 10·31-s − 10·36-s − 20·41-s + 8·44-s + 9·49-s − 18·59-s − 34·61-s − 4·64-s + 24·76-s − 2·79-s + 4·81-s + 44·89-s − 20·99-s − 30·101-s + 44·109-s + 8·116-s − 8·121-s + 20·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s + 5/3·9-s − 1.20·11-s + 3/4·16-s − 2.75·19-s − 0.742·29-s − 1.79·31-s − 5/3·36-s − 3.12·41-s + 1.20·44-s + 9/7·49-s − 2.34·59-s − 4.35·61-s − 1/2·64-s + 2.75·76-s − 0.225·79-s + 4/9·81-s + 4.66·89-s − 2.01·99-s − 2.98·101-s + 4.21·109-s + 0.742·116-s − 0.727·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5720572401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5720572401\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 7 p T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 9 T^{2} + 37 T^{4} - 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 81 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 321 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 61 T^{2} + 1829 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 88 T^{2} + 4206 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 161 T^{2} + 10149 T^{4} - 161 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 149 T^{2} + 10905 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 9 T + 109 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 17 T + 165 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 14230 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 45 T^{2} - 2567 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + T + 129 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 28862 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 22 T + 286 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 28881 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.73498678801770435440978168179, −6.71475137899833044835584656319, −6.27648239971479578216313385115, −6.23037465388274450395052809983, −5.81860893541755335968935856479, −5.77008806569789362088313519445, −5.56463385151346659304099688097, −4.93636456837988367681870211964, −4.90232303695332888213484002994, −4.82344170162601018143572561715, −4.76402768615726353439358678894, −4.40830671801111321869251410726, −3.97573919137954430439863296118, −3.86455569611502057192139370924, −3.72912274742285476047244782898, −3.53108349246535564761866250355, −3.04975860550152159782759450250, −2.71698539084578991462752636682, −2.67922422659027618745887904629, −2.00474466670968750336781631909, −1.76028422211911830902300797787, −1.59044157146350117275570730785, −1.56828002747102530951880173100, −0.54303561979431377975275011123, −0.22309634533403511021705049936,
0.22309634533403511021705049936, 0.54303561979431377975275011123, 1.56828002747102530951880173100, 1.59044157146350117275570730785, 1.76028422211911830902300797787, 2.00474466670968750336781631909, 2.67922422659027618745887904629, 2.71698539084578991462752636682, 3.04975860550152159782759450250, 3.53108349246535564761866250355, 3.72912274742285476047244782898, 3.86455569611502057192139370924, 3.97573919137954430439863296118, 4.40830671801111321869251410726, 4.76402768615726353439358678894, 4.82344170162601018143572561715, 4.90232303695332888213484002994, 4.93636456837988367681870211964, 5.56463385151346659304099688097, 5.77008806569789362088313519445, 5.81860893541755335968935856479, 6.23037465388274450395052809983, 6.27648239971479578216313385115, 6.71475137899833044835584656319, 6.73498678801770435440978168179
