| L(s) = 1 | − 5·3-s + 4-s + 13·9-s − 5·12-s − 13-s − 3·16-s − 17-s + 3·23-s + 7·25-s − 20·27-s − 11·29-s + 13·36-s + 5·39-s + 8·43-s + 15·48-s + 49-s + 5·51-s − 52-s − 14·53-s − 5·61-s − 7·64-s − 68-s − 15·69-s − 35·75-s − 10·79-s + 10·81-s + 55·87-s + ⋯ |
| L(s) = 1 | − 2.88·3-s + 1/2·4-s + 13/3·9-s − 1.44·12-s − 0.277·13-s − 3/4·16-s − 0.242·17-s + 0.625·23-s + 7/5·25-s − 3.84·27-s − 2.04·29-s + 13/6·36-s + 0.800·39-s + 1.21·43-s + 2.16·48-s + 1/7·49-s + 0.700·51-s − 0.138·52-s − 1.92·53-s − 0.640·61-s − 7/8·64-s − 0.121·68-s − 1.80·69-s − 4.04·75-s − 1.12·79-s + 10/9·81-s + 5.89·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37349 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37349 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | $C_1$ | \( 1 + T \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
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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
−10.59044501959644866767132722404, −9.674844609797841846358054299712, −9.277417665407585805027947316521, −8.576914825215962727607942877634, −7.51341240600202576695190822198, −7.18880659578063013328516195009, −6.64851863238724920504786965514, −6.16197865789512764318391536105, −5.77797745461673806148537077881, −5.00841943187520197105472149545, −4.85289720084871064985022363880, −3.98139969512858312145116457368, −2.71249778982484038202881000224, −1.37955483692025509929371250557, 0,
1.37955483692025509929371250557, 2.71249778982484038202881000224, 3.98139969512858312145116457368, 4.85289720084871064985022363880, 5.00841943187520197105472149545, 5.77797745461673806148537077881, 6.16197865789512764318391536105, 6.64851863238724920504786965514, 7.18880659578063013328516195009, 7.51341240600202576695190822198, 8.576914825215962727607942877634, 9.277417665407585805027947316521, 9.674844609797841846358054299712, 10.59044501959644866767132722404
