L(s) = 1 | + 4-s + 6·7-s − 2·13-s − 3·16-s − 19-s + 4·25-s + 6·28-s + 6·31-s + 4·43-s + 14·49-s − 2·52-s + 8·61-s − 7·64-s + 6·67-s − 76-s − 4·79-s − 12·91-s + 18·97-s + 4·100-s − 8·103-s + 18·109-s − 18·112-s − 14·121-s + 6·124-s + 127-s + 131-s − 6·133-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 2.26·7-s − 0.554·13-s − 3/4·16-s − 0.229·19-s + 4/5·25-s + 1.13·28-s + 1.07·31-s + 0.609·43-s + 2·49-s − 0.277·52-s + 1.02·61-s − 7/8·64-s + 0.733·67-s − 0.114·76-s − 0.450·79-s − 1.25·91-s + 1.82·97-s + 2/5·100-s − 0.788·103-s + 1.72·109-s − 1.70·112-s − 1.27·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s − 0.520·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260091 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260091 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.606698220\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.606698220\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.691558074608534275327050692828, −8.462892564697675846675425773749, −8.023538540642562152790932836286, −7.43158152605048932532567630543, −7.25673784161257788031657543738, −6.52582150413373997413341317595, −6.14222157771019265223374718121, −5.25732862181974178709149682753, −5.04109205807352546423433647398, −4.50740908059770894750540572941, −4.10900493280774529987881316352, −3.09448184731489353256442152888, −2.35051797167753395964823441603, −1.92617471838960901505654878375, −1.04690143485382768458934991120,
1.04690143485382768458934991120, 1.92617471838960901505654878375, 2.35051797167753395964823441603, 3.09448184731489353256442152888, 4.10900493280774529987881316352, 4.50740908059770894750540572941, 5.04109205807352546423433647398, 5.25732862181974178709149682753, 6.14222157771019265223374718121, 6.52582150413373997413341317595, 7.25673784161257788031657543738, 7.43158152605048932532567630543, 8.023538540642562152790932836286, 8.462892564697675846675425773749, 8.691558074608534275327050692828