| L(s) = 1 | + 4·7-s + 2·9-s − 12·17-s + 12·23-s − 25-s + 8·31-s − 12·41-s + 12·47-s − 2·49-s + 8·63-s − 24·71-s − 4·73-s − 16·79-s − 5·81-s + 12·89-s + 4·97-s + 28·103-s − 12·113-s − 48·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2/3·9-s − 2.91·17-s + 2.50·23-s − 1/5·25-s + 1.43·31-s − 1.87·41-s + 1.75·47-s − 2/7·49-s + 1.00·63-s − 2.84·71-s − 0.468·73-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 0.406·97-s + 2.75·103-s − 1.12·113-s − 4.40·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.585913676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.585913676\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.20052898539242611397909731122, −9.148085412675813891936646785210, −8.959129842932469534164806714545, −8.801834284339849304294155559021, −8.385500135164812514756384431203, −7.78517028695477112170101699191, −7.44710099286074170155524085338, −6.89851601602176498797041632823, −6.76048083043038491165210125637, −6.27477430921408277322119933101, −5.59836186515787395494516653704, −5.04171054891648705236366295193, −4.69285146601263802022539723429, −4.42029860877984215347666245476, −4.11355726121156523173933348244, −3.03500349830528548119990947945, −2.81019808355718984241074660766, −1.81280180084146899617098350177, −1.71448922623350348090211524349, −0.68873071227057691255821670662,
0.68873071227057691255821670662, 1.71448922623350348090211524349, 1.81280180084146899617098350177, 2.81019808355718984241074660766, 3.03500349830528548119990947945, 4.11355726121156523173933348244, 4.42029860877984215347666245476, 4.69285146601263802022539723429, 5.04171054891648705236366295193, 5.59836186515787395494516653704, 6.27477430921408277322119933101, 6.76048083043038491165210125637, 6.89851601602176498797041632823, 7.44710099286074170155524085338, 7.78517028695477112170101699191, 8.385500135164812514756384431203, 8.801834284339849304294155559021, 8.959129842932469534164806714545, 9.148085412675813891936646785210, 10.20052898539242611397909731122
