| L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 4·13-s − 8·21-s + 25-s + 4·27-s − 8·31-s + 12·37-s + 8·39-s − 4·43-s − 2·49-s + 12·61-s + 4·63-s − 12·67-s − 12·73-s − 2·75-s + 16·79-s − 11·81-s − 16·91-s + 16·93-s − 12·97-s − 4·103-s − 20·109-s − 24·111-s − 4·117-s + 10·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s − 1.10·13-s − 1.74·21-s + 1/5·25-s + 0.769·27-s − 1.43·31-s + 1.97·37-s + 1.28·39-s − 0.609·43-s − 2/7·49-s + 1.53·61-s + 0.503·63-s − 1.46·67-s − 1.40·73-s − 0.230·75-s + 1.80·79-s − 1.22·81-s − 1.67·91-s + 1.65·93-s − 1.21·97-s − 0.394·103-s − 1.91·109-s − 2.27·111-s − 0.369·117-s + 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.86705417337960620856668217142, −7.52087961917681852705170049172, −7.07718025517805270139079078974, −6.60021656167659173349229803577, −6.07798718905310210884823128498, −5.60789325607463077823008353904, −5.19659620613578902877270285781, −4.85959012122126862184020046215, −4.46801282759482952851937133707, −3.93992690783982909620590989811, −3.08497662827967458664982166905, −2.45689282980853636600855306564, −1.79067373657145419734549268247, −1.08406870694938546769851272715, 0,
1.08406870694938546769851272715, 1.79067373657145419734549268247, 2.45689282980853636600855306564, 3.08497662827967458664982166905, 3.93992690783982909620590989811, 4.46801282759482952851937133707, 4.85959012122126862184020046215, 5.19659620613578902877270285781, 5.60789325607463077823008353904, 6.07798718905310210884823128498, 6.60021656167659173349229803577, 7.07718025517805270139079078974, 7.52087961917681852705170049172, 7.86705417337960620856668217142
