| L(s) = 1 | + 4·13-s + 4·25-s + 4·37-s + 22·49-s − 44·61-s − 4·73-s − 52·97-s + 40·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.10·13-s + 4/5·25-s + 0.657·37-s + 22/7·49-s − 5.63·61-s − 0.468·73-s − 5.27·97-s + 3.83·109-s + 4/11·121-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 − 3.23·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.726203847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.726203847\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 155 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) |
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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
−6.78223786931276240965669413526, −6.41317143934561923967195591340, −6.02076275666442767918391649765, −5.93789057309032941434991731186, −5.88042799431600256309065630045, −5.68931335684947911323347140921, −5.53748269821118256482051352962, −5.01076945464448825643268020962, −4.82231514509249154918294601857, −4.69544931676827314188699924549, −4.45147334543587499772078685744, −4.25742208313532796767268826147, −4.02627562760468768915380182217, −3.70452678906309732307841892489, −3.45274099738324577098361050550, −3.30478726462640934107527356211, −2.94848128629919209385026601224, −2.71807492369847361217513547802, −2.44576792246002894164052230728, −2.31246860715987068398698176048, −1.63517492241153718724265220868, −1.44622892923630363541304710342, −1.34945318674062353368035485073, −0.789782212133105080676238111218, −0.33457710453920027028744614369,
0.33457710453920027028744614369, 0.789782212133105080676238111218, 1.34945318674062353368035485073, 1.44622892923630363541304710342, 1.63517492241153718724265220868, 2.31246860715987068398698176048, 2.44576792246002894164052230728, 2.71807492369847361217513547802, 2.94848128629919209385026601224, 3.30478726462640934107527356211, 3.45274099738324577098361050550, 3.70452678906309732307841892489, 4.02627562760468768915380182217, 4.25742208313532796767268826147, 4.45147334543587499772078685744, 4.69544931676827314188699924549, 4.82231514509249154918294601857, 5.01076945464448825643268020962, 5.53748269821118256482051352962, 5.68931335684947911323347140921, 5.88042799431600256309065630045, 5.93789057309032941434991731186, 6.02076275666442767918391649765, 6.41317143934561923967195591340, 6.78223786931276240965669413526
