| L(s) = 1 | + 3·2-s + 5·4-s + 5·8-s + 9-s + 11-s + 4·13-s + 16-s + 3·18-s + 3·22-s − 2·23-s − 4·25-s + 12·26-s − 2·29-s − 7·32-s + 5·36-s + 18·41-s + 5·44-s − 6·46-s + 16·47-s + 9·49-s − 12·50-s + 20·52-s − 6·58-s − 15·64-s + 5·72-s − 6·79-s − 8·81-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 5/2·4-s + 1.76·8-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1/4·16-s + 0.707·18-s + 0.639·22-s − 0.417·23-s − 4/5·25-s + 2.35·26-s − 0.371·29-s − 1.23·32-s + 5/6·36-s + 2.81·41-s + 0.753·44-s − 0.884·46-s + 2.33·47-s + 9/7·49-s − 1.69·50-s + 2.77·52-s − 0.787·58-s − 1.87·64-s + 0.589·72-s − 0.675·79-s − 8/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 436810 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 436810 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.801876264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.801876264\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - p T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 145 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 171 T^{2} + p^{2} T^{4} \) |
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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
−8.718176002837950505871583439736, −7.906829575138703699033242264995, −7.44643449586441236244994083305, −7.11497559558878777516552373423, −6.47410146988496011644991874585, −5.93280151001694828437782638868, −5.81871303885319435219014786705, −5.40067292478301779812067875475, −4.54677029527379988477977723498, −4.15395232164980027145316971087, −3.95452389521112325388768944088, −3.37094577131521141905762543496, −2.60258557018662325943020943419, −2.15408890433173940057711794976, −1.08577083337626308906712774892,
1.08577083337626308906712774892, 2.15408890433173940057711794976, 2.60258557018662325943020943419, 3.37094577131521141905762543496, 3.95452389521112325388768944088, 4.15395232164980027145316971087, 4.54677029527379988477977723498, 5.40067292478301779812067875475, 5.81871303885319435219014786705, 5.93280151001694828437782638868, 6.47410146988496011644991874585, 7.11497559558878777516552373423, 7.44643449586441236244994083305, 7.906829575138703699033242264995, 8.718176002837950505871583439736
