| L(s) = 1 | + 5-s − 4·7-s + 4·11-s + 17-s − 19-s − 8·23-s − 6·29-s − 6·31-s − 4·35-s + 8·37-s + 4·41-s − 12·43-s + 47-s − 2·49-s + 5·53-s + 4·55-s + 10·59-s − 14·61-s + 6·67-s + 8·71-s + 2·73-s − 16·77-s + 8·79-s − 22·83-s + 85-s − 2·89-s − 95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.20·11-s + 0.242·17-s − 0.229·19-s − 1.66·23-s − 1.11·29-s − 1.07·31-s − 0.676·35-s + 1.31·37-s + 0.624·41-s − 1.82·43-s + 0.145·47-s − 2/7·49-s + 0.686·53-s + 0.539·55-s + 1.30·59-s − 1.79·61-s + 0.733·67-s + 0.949·71-s + 0.234·73-s − 1.82·77-s + 0.900·79-s − 2.41·83-s + 0.108·85-s − 0.211·89-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11696400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5460354644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5460354644\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.065028658824763075778000729500, −8.224950933800892271472740523841, −8.176081961855018222762484544990, −7.73603517787377274817235536144, −7.16869589826159560793796988323, −6.81577837055858474109206676781, −6.48339352041485547963870987568, −6.28745143132961131268773339505, −5.88688253782750277636231598460, −5.32415670449953869662197738531, −5.29289955413278947491306579184, −4.32950426500267241699730581278, −4.06003920095048170217922718530, −3.68318918364032531025270630202, −3.45379306235133050218649787537, −2.61178713778954493186889770547, −2.49721895894043358065231138570, −1.58275988769598904157815770580, −1.38795199095008202113356824690, −0.21852618611805249580193059099,
0.21852618611805249580193059099, 1.38795199095008202113356824690, 1.58275988769598904157815770580, 2.49721895894043358065231138570, 2.61178713778954493186889770547, 3.45379306235133050218649787537, 3.68318918364032531025270630202, 4.06003920095048170217922718530, 4.32950426500267241699730581278, 5.29289955413278947491306579184, 5.32415670449953869662197738531, 5.88688253782750277636231598460, 6.28745143132961131268773339505, 6.48339352041485547963870987568, 6.81577837055858474109206676781, 7.16869589826159560793796988323, 7.73603517787377274817235536144, 8.176081961855018222762484544990, 8.224950933800892271472740523841, 9.065028658824763075778000729500
