| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 9-s + 14-s + 16-s + 18-s + 3·23-s − 4·25-s + 28-s + 7·31-s + 32-s + 36-s − 6·41-s + 3·46-s − 3·47-s + 7·49-s − 4·50-s + 56-s + 7·62-s + 63-s + 64-s + 9·71-s + 72-s − 11·73-s + 79-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.625·23-s − 4/5·25-s + 0.188·28-s + 1.25·31-s + 0.176·32-s + 1/6·36-s − 0.937·41-s + 0.442·46-s − 0.437·47-s + 49-s − 0.565·50-s + 0.133·56-s + 0.889·62-s + 0.125·63-s + 1/8·64-s + 1.06·71-s + 0.117·72-s − 1.28·73-s + 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.111163643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.111163643\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.42367481099616068122609563986, −9.850176488145126585278785263807, −9.473333045066788301581458681440, −8.617060546894272028894031477544, −8.234815281550343853648207014470, −7.66227293224260701056211036325, −6.92722922280120151656471211130, −6.68831323895571547724810748670, −5.76794768375248488375283321368, −5.40486828149287321724051390604, −4.56983283733783865990546444238, −4.18000462007726669866019308591, −3.29262758395902131285047238697, −2.51598487599313506037092043707, −1.45299147381499260458508100761,
1.45299147381499260458508100761, 2.51598487599313506037092043707, 3.29262758395902131285047238697, 4.18000462007726669866019308591, 4.56983283733783865990546444238, 5.40486828149287321724051390604, 5.76794768375248488375283321368, 6.68831323895571547724810748670, 6.92722922280120151656471211130, 7.66227293224260701056211036325, 8.234815281550343853648207014470, 8.617060546894272028894031477544, 9.473333045066788301581458681440, 9.850176488145126585278785263807, 10.42367481099616068122609563986
