| L(s) = 1 | − 8·16-s − 2·25-s − 16·43-s + 2·49-s + 52·61-s + 52·79-s + 40·103-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2·16-s − 2/5·25-s − 2.43·43-s + 2/7·49-s + 6.65·61-s + 5.85·79-s + 3.94·103-s + 2.36·121-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 − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.353841925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.353841925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 21 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 39 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 169 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 181 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.81293144968649727380666148945, −7.35449638481312468048724572557, −7.03708597777763116281173647207, −7.03152230156419155182142912956, −6.94391595049713855404006931951, −6.48528091341689320842539822025, −6.33746068979094599465136030378, −6.13476236443244883773583525109, −5.87180431971600677317119580184, −5.33320711136710335634204686467, −5.15547989339809802879574550380, −5.12302110017609101935014833741, −4.79420707122433485941673250125, −4.56971035817774309661104182808, −4.12506382401174225492552797058, −3.76840870973028695816046339357, −3.75120582976975299775343947977, −3.38007843452385266801304455355, −3.06659558239422310515133068402, −2.50548538284112622547594275756, −2.17691516096619175552700088884, −2.03689542962607304281943648936, −1.85551754050527559636569970945, −0.70107886146530611510706438638, −0.69912277573216465176782182821,
0.69912277573216465176782182821, 0.70107886146530611510706438638, 1.85551754050527559636569970945, 2.03689542962607304281943648936, 2.17691516096619175552700088884, 2.50548538284112622547594275756, 3.06659558239422310515133068402, 3.38007843452385266801304455355, 3.75120582976975299775343947977, 3.76840870973028695816046339357, 4.12506382401174225492552797058, 4.56971035817774309661104182808, 4.79420707122433485941673250125, 5.12302110017609101935014833741, 5.15547989339809802879574550380, 5.33320711136710335634204686467, 5.87180431971600677317119580184, 6.13476236443244883773583525109, 6.33746068979094599465136030378, 6.48528091341689320842539822025, 6.94391595049713855404006931951, 7.03152230156419155182142912956, 7.03708597777763116281173647207, 7.35449638481312468048724572557, 7.81293144968649727380666148945
