| L(s) = 1 | − 378·5-s + 832·7-s − 2.48e3·11-s − 1.48e4·13-s + 4.46e4·17-s − 3.26e4·19-s − 1.15e5·23-s + 7.81e4·25-s + 1.57e5·29-s + 1.64e4·31-s − 3.14e5·35-s − 2.98e5·37-s − 2.41e5·41-s + 4.43e5·43-s + 9.22e5·47-s + 8.23e5·49-s + 1.39e6·53-s + 9.38e5·55-s + 8.70e5·59-s − 2.06e6·61-s + 5.62e6·65-s + 1.68e6·67-s + 2.14e6·71-s − 4.80e6·73-s − 2.06e6·77-s − 2.30e6·79-s + 4.70e6·83-s + ⋯ |
| L(s) = 1 | − 1.35·5-s + 0.916·7-s − 0.562·11-s − 1.87·13-s + 2.20·17-s − 1.09·19-s − 1.97·23-s + 25-s + 1.19·29-s + 0.0992·31-s − 1.23·35-s − 0.968·37-s − 0.546·41-s + 0.850·43-s + 1.29·47-s + 49-s + 1.28·53-s + 0.760·55-s + 0.551·59-s − 1.16·61-s + 2.53·65-s + 0.682·67-s + 0.709·71-s − 1.44·73-s − 0.515·77-s − 0.525·79-s + 0.903·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9709574418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9709574418\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 378 T + 64759 T^{2} + 378 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 832 T - 131319 T^{2} - 832 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2484 T - 13316915 T^{2} + 2484 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14870 T + 158368383 T^{2} + 14870 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 22302 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 16300 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 115128 T + 9849630937 T^{2} + 115128 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 157086 T + 7426135087 T^{2} - 157086 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 16456 T - 27241814175 T^{2} - 16456 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 149266 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 241110 T - 136620241781 T^{2} + 241110 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 443188 T - 75403007763 T^{2} - 443188 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 922752 T + 344848133041 T^{2} - 922752 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 697626 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 870156 T - 1731480020483 T^{2} - 870156 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2067062 T + 1130002475823 T^{2} + 2067062 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1680748 T - 3235797765819 T^{2} - 1680748 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1070280 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2403334 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 2301512 T - 13906951500015 T^{2} + 2301512 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4708692 T - 4964270638763 T^{2} - 4708692 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4143690 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1622974 T - 78164239873437 T^{2} - 1622974 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.17143078664331371873261258954, −10.17416153511107116221470412923, −10.08704648629052126749073309964, −9.132212952708728994333764396386, −8.603681962579512192701814746879, −7.989431575931875965295574874644, −7.982140813027195825024858396223, −7.27600616260546647479192309962, −7.25210554981747967259510905659, −6.23576104879415417638753716939, −5.65504124479702129337373074730, −5.09489912308501469375724860760, −4.73528631260914308434547131703, −3.98216788421601176640971561772, −3.81968226313702236197496097120, −2.79919758290731311643916226964, −2.44904203169101438412367709858, −1.69117234783224741947129226815, −0.868937229216328879601754603689, −0.26465259115138726319495930661,
0.26465259115138726319495930661, 0.868937229216328879601754603689, 1.69117234783224741947129226815, 2.44904203169101438412367709858, 2.79919758290731311643916226964, 3.81968226313702236197496097120, 3.98216788421601176640971561772, 4.73528631260914308434547131703, 5.09489912308501469375724860760, 5.65504124479702129337373074730, 6.23576104879415417638753716939, 7.25210554981747967259510905659, 7.27600616260546647479192309962, 7.982140813027195825024858396223, 7.989431575931875965295574874644, 8.603681962579512192701814746879, 9.132212952708728994333764396386, 10.08704648629052126749073309964, 10.17416153511107116221470412923, 11.17143078664331371873261258954
