| L(s) = 1 | + 2·9-s + 12·11-s + 12·23-s + 10·25-s − 4·37-s − 4·49-s − 44·67-s − 36·71-s − 15·81-s + 24·99-s − 12·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 3.61·11-s + 2.50·23-s + 2·25-s − 0.657·37-s − 4/7·49-s − 5.37·67-s − 4.27·71-s − 5/3·81-s + 2.41·99-s − 1.12·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.978599477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.978599477\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 136 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 173 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 149 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.97792650558364186517294018599, −6.82531273143171045690050860177, −6.55066269547999907306273893163, −6.28617455634697364319278079491, −6.10756877235843440314977399119, −6.01575067939305668588878806518, −5.66347128252810403950060603700, −5.42196545087892792511555714865, −4.95873334135115377514803632250, −4.92412785900335810059070846444, −4.51331879040944607046178919494, −4.50440851723466944615809731540, −4.18545337933987888266949729514, −4.07996347798920515985039909355, −3.75256782382810686277329615839, −3.19537056146018256552951646254, −3.18366011834878477684170431743, −3.11677516092445479752027791405, −2.70516626812986673353278327861, −2.34658427216139065751626210794, −1.56341748315225492556090834085, −1.43929644792923261774976075637, −1.32338818904276140989184174216, −1.27982278696716040161900821061, −0.45582375148949135865595755256,
0.45582375148949135865595755256, 1.27982278696716040161900821061, 1.32338818904276140989184174216, 1.43929644792923261774976075637, 1.56341748315225492556090834085, 2.34658427216139065751626210794, 2.70516626812986673353278327861, 3.11677516092445479752027791405, 3.18366011834878477684170431743, 3.19537056146018256552951646254, 3.75256782382810686277329615839, 4.07996347798920515985039909355, 4.18545337933987888266949729514, 4.50440851723466944615809731540, 4.51331879040944607046178919494, 4.92412785900335810059070846444, 4.95873334135115377514803632250, 5.42196545087892792511555714865, 5.66347128252810403950060603700, 6.01575067939305668588878806518, 6.10756877235843440314977399119, 6.28617455634697364319278079491, 6.55066269547999907306273893163, 6.82531273143171045690050860177, 6.97792650558364186517294018599
