| L(s) = 1 | + 2·7-s + 2·9-s − 25-s − 4·47-s + 3·49-s + 4·63-s + 3·81-s + 4·103-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2·7-s + 2·9-s − 25-s − 4·47-s + 3·49-s + 4·63-s + 3·81-s + 4·103-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.992737085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992737085\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$ | \( ( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.631300000662440740279224043059, −8.981570027274461453613256891649, −8.464676696127892362397824252305, −8.295089341788594883270856524188, −7.76597902520911339240629675434, −7.56768697351250667926202147923, −7.27324706375228574405779357587, −6.76398526671903995521756419248, −6.31653561507417524891879996530, −5.96512522636659351344432566769, −5.19694415713145036871169478614, −4.99077114415019405319091170088, −4.59899074737076501756232660675, −4.38703423884987908501704823677, −3.58676174791693956848194821155, −3.58194766541455781027288932579, −2.46768091778759055559457251370, −1.96421149648571824914392663660, −1.56767383836046793110270484746, −1.16363538154661757476284320733,
1.16363538154661757476284320733, 1.56767383836046793110270484746, 1.96421149648571824914392663660, 2.46768091778759055559457251370, 3.58194766541455781027288932579, 3.58676174791693956848194821155, 4.38703423884987908501704823677, 4.59899074737076501756232660675, 4.99077114415019405319091170088, 5.19694415713145036871169478614, 5.96512522636659351344432566769, 6.31653561507417524891879996530, 6.76398526671903995521756419248, 7.27324706375228574405779357587, 7.56768697351250667926202147923, 7.76597902520911339240629675434, 8.295089341788594883270856524188, 8.464676696127892362397824252305, 8.981570027274461453613256891649, 9.631300000662440740279224043059
