| L(s) = 1 | − 3-s + 3·7-s − 2·9-s − 5·13-s − 7·19-s − 3·21-s − 7·25-s + 5·27-s + 4·31-s − 9·37-s + 5·39-s − 2·43-s − 7·49-s + 7·57-s − 11·61-s − 6·63-s + 10·67-s + 13·73-s + 7·75-s − 9·79-s + 81-s − 15·91-s − 4·93-s − 2·97-s − 25·103-s − 9·109-s + 9·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s − 1.38·13-s − 1.60·19-s − 0.654·21-s − 7/5·25-s + 0.962·27-s + 0.718·31-s − 1.47·37-s + 0.800·39-s − 0.304·43-s − 49-s + 0.927·57-s − 1.40·61-s − 0.755·63-s + 1.22·67-s + 1.52·73-s + 0.808·75-s − 1.01·79-s + 1/9·81-s − 1.57·91-s − 0.414·93-s − 0.203·97-s − 2.46·103-s − 0.862·109-s + 0.854·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.733842098132795413336316861401, −9.012720569710318579302723316031, −8.440319378902538863652513828177, −8.079743453939254536273309811449, −7.69851372523135654882469391175, −6.81966604208790228380394133191, −6.56895659176718719435570656418, −5.74591405816308452711486557257, −5.33371054209633628102910099290, −4.70033648388387210857421129256, −4.35438128134365865488605170476, −3.36269409628234175254002555523, −2.41170878034698080184969794152, −1.77431610907885565269488703309, 0,
1.77431610907885565269488703309, 2.41170878034698080184969794152, 3.36269409628234175254002555523, 4.35438128134365865488605170476, 4.70033648388387210857421129256, 5.33371054209633628102910099290, 5.74591405816308452711486557257, 6.56895659176718719435570656418, 6.81966604208790228380394133191, 7.69851372523135654882469391175, 8.079743453939254536273309811449, 8.440319378902538863652513828177, 9.012720569710318579302723316031, 9.733842098132795413336316861401
