L(s) = 1 | + 3·3-s + 6·9-s + 5·11-s + 7·19-s + 25-s + 9·27-s + 15·33-s − 11·41-s + 2·43-s − 10·49-s + 21·57-s − 59-s + 16·67-s + 9·73-s + 3·75-s + 9·81-s + 3·83-s − 13·89-s + 2·97-s + 30·99-s + 13·107-s − 10·113-s − 121-s − 33·123-s + 127-s + 6·129-s + 131-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 2·9-s + 1.50·11-s + 1.60·19-s + 1/5·25-s + 1.73·27-s + 2.61·33-s − 1.71·41-s + 0.304·43-s − 1.42·49-s + 2.78·57-s − 0.130·59-s + 1.95·67-s + 1.05·73-s + 0.346·75-s + 81-s + 0.329·83-s − 1.37·89-s + 0.203·97-s + 3.01·99-s + 1.25·107-s − 0.940·113-s − 0.0909·121-s − 2.97·123-s + 0.0887·127-s + 0.528·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.332035378\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.332035378\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−7.987188563380098501084118686166, −7.72225338619090257059299808563, −7.09850200537895055189195754187, −6.74652524675670337148360693510, −6.56320886023603469249088358892, −5.75288045395424730543507259187, −5.23459261483295082973709731666, −4.73027611457244648930347495110, −4.17007192722853740194518967768, −3.66132965534643201626662584217, −3.34259543456536734900594559208, −2.93586245278622071576352721636, −2.14730059509799811767189732407, −1.61543855451612350693233033437, −1.00597017015718147002552005790,
1.00597017015718147002552005790, 1.61543855451612350693233033437, 2.14730059509799811767189732407, 2.93586245278622071576352721636, 3.34259543456536734900594559208, 3.66132965534643201626662584217, 4.17007192722853740194518967768, 4.73027611457244648930347495110, 5.23459261483295082973709731666, 5.75288045395424730543507259187, 6.56320886023603469249088358892, 6.74652524675670337148360693510, 7.09850200537895055189195754187, 7.72225338619090257059299808563, 7.987188563380098501084118686166