| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 2·9-s − 6·11-s − 2·13-s − 2·14-s + 16-s + 2·18-s − 6·22-s − 4·25-s − 2·26-s − 2·28-s + 6·31-s + 32-s + 2·36-s − 8·41-s − 6·44-s − 10·49-s − 4·50-s − 2·52-s − 6·53-s − 2·56-s − 16·61-s + 6·62-s − 4·63-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 2/3·9-s − 1.80·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s + 0.471·18-s − 1.27·22-s − 4/5·25-s − 0.392·26-s − 0.377·28-s + 1.07·31-s + 0.176·32-s + 1/3·36-s − 1.24·41-s − 0.904·44-s − 1.42·49-s − 0.565·50-s − 0.277·52-s − 0.824·53-s − 0.267·56-s − 2.04·61-s + 0.762·62-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 204304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 204304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 113 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.788815138471885023988167837781, −8.145585223353684139184758662884, −7.86453140337875596672122410605, −7.37456268740768612871238353725, −6.87693994519191936158546934810, −6.35876366994336793845045705403, −5.87880191616178278555995682376, −5.31629747230410132333713822502, −4.70015150394815352026541727879, −4.50195428200827899320596033445, −3.50214828449880324343422352520, −3.07316307119561726841246213112, −2.47204509350364564322118684422, −1.65099944678956700174546558154, 0,
1.65099944678956700174546558154, 2.47204509350364564322118684422, 3.07316307119561726841246213112, 3.50214828449880324343422352520, 4.50195428200827899320596033445, 4.70015150394815352026541727879, 5.31629747230410132333713822502, 5.87880191616178278555995682376, 6.35876366994336793845045705403, 6.87693994519191936158546934810, 7.37456268740768612871238353725, 7.86453140337875596672122410605, 8.145585223353684139184758662884, 8.788815138471885023988167837781
