| L(s) = 1 | − 4-s + 16-s + 2·25-s + 4·43-s − 64-s + 4·67-s − 2·100-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 − 4·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s + 2·25-s + 4·43-s − 64-s + 4·67-s − 2·100-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 − 4·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.256126637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256126637\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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
−8.899361819602058303556580214097, −8.725541625605721927706512258917, −8.124673474241451557566521976661, −7.964367688141004545372412970588, −7.52776675909715434508590690277, −7.08637248517300635981022897677, −6.75589416357485867414871967653, −6.31777272154704781262586250683, −5.88824008691254355784721637350, −5.41520702284285638226380296923, −5.22450797416426672194843837321, −4.69687866381282804432004504054, −4.39743715455457080207742659710, −3.76621642528939160337731498578, −3.76421719505902442970045466875, −2.91982300187330564139886741939, −2.61576119687548318090402244022, −2.08613502998531756224438738549, −1.08731138611794378212789298646, −0.843981798799303935556929761975,
0.843981798799303935556929761975, 1.08731138611794378212789298646, 2.08613502998531756224438738549, 2.61576119687548318090402244022, 2.91982300187330564139886741939, 3.76421719505902442970045466875, 3.76621642528939160337731498578, 4.39743715455457080207742659710, 4.69687866381282804432004504054, 5.22450797416426672194843837321, 5.41520702284285638226380296923, 5.88824008691254355784721637350, 6.31777272154704781262586250683, 6.75589416357485867414871967653, 7.08637248517300635981022897677, 7.52776675909715434508590690277, 7.964367688141004545372412970588, 8.124673474241451557566521976661, 8.725541625605721927706512258917, 8.899361819602058303556580214097
