| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 4·9-s + 10-s + 7·11-s + 13-s + 4·14-s + 16-s − 4·18-s − 20-s − 7·22-s − 4·25-s − 26-s − 4·28-s + 5·31-s − 32-s + 4·35-s + 4·36-s + 40-s − 3·43-s + 7·44-s − 4·45-s + 9·47-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 4/3·9-s + 0.316·10-s + 2.11·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.942·18-s − 0.223·20-s − 1.49·22-s − 4/5·25-s − 0.196·26-s − 0.755·28-s + 0.898·31-s − 0.176·32-s + 0.676·35-s + 2/3·36-s + 0.158·40-s − 0.457·43-s + 1.05·44-s − 0.596·45-s + 1.31·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.127663788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.127663788\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
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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
−9.263740080514530709283222910006, −9.023027853257041742050712186909, −8.270468553290966752863252888305, −7.911521286990417869147426559673, −7.11045502477438807522114766800, −6.73819554778030425119584218135, −6.67880368733453725411842430338, −6.02135939158773646944415241356, −5.37412936261027250493200161140, −4.24991787558228170664864268717, −4.00489604673684129900823124611, −3.57338086754239314689177595609, −2.70863923264644426901342286294, −1.68317790647196074299899078083, −0.853742979617782077749245083370,
0.853742979617782077749245083370, 1.68317790647196074299899078083, 2.70863923264644426901342286294, 3.57338086754239314689177595609, 4.00489604673684129900823124611, 4.24991787558228170664864268717, 5.37412936261027250493200161140, 6.02135939158773646944415241356, 6.67880368733453725411842430338, 6.73819554778030425119584218135, 7.11045502477438807522114766800, 7.911521286990417869147426559673, 8.270468553290966752863252888305, 9.023027853257041742050712186909, 9.263740080514530709283222910006
