| L(s) = 1 | − 4-s − 4·5-s − 9-s − 10·11-s + 16-s + 14·19-s + 4·20-s + 11·25-s + 12·31-s + 36-s + 18·41-s + 10·44-s + 4·45-s + 40·55-s + 8·59-s + 4·61-s − 64-s − 4·71-s − 14·76-s + 28·79-s − 4·80-s + 81-s + 20·89-s − 56·95-s + 10·99-s − 11·100-s − 16·101-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s − 1/3·9-s − 3.01·11-s + 1/4·16-s + 3.21·19-s + 0.894·20-s + 11/5·25-s + 2.15·31-s + 1/6·36-s + 2.81·41-s + 1.50·44-s + 0.596·45-s + 5.39·55-s + 1.04·59-s + 0.512·61-s − 1/8·64-s − 0.474·71-s − 1.60·76-s + 3.15·79-s − 0.447·80-s + 1/9·81-s + 2.11·89-s − 5.74·95-s + 1.00·99-s − 1.09·100-s − 1.59·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.184198327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184198327\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.523016099171869295072700801870, −9.506858341875896690407733367347, −8.753458223904739243849774299404, −8.324780208928235940542224213421, −7.966098742367669350050663928217, −7.79255165723859507121450273367, −7.36332304463235693496451944900, −7.35894510818848211701162857017, −6.47508992198743382441741548276, −5.83076933706513894768780898637, −5.44836009546409179146669427874, −5.01934241822842419254120534218, −4.78853758723080216506020188348, −4.32223120364492829701001689555, −3.43212948861950038759447224791, −3.38005002259004427887055595411, −2.65957645536533251781486224490, −2.48684151330262709961092652433, −0.75155539483712145679195271728, −0.71703134141693045798973665784,
0.71703134141693045798973665784, 0.75155539483712145679195271728, 2.48684151330262709961092652433, 2.65957645536533251781486224490, 3.38005002259004427887055595411, 3.43212948861950038759447224791, 4.32223120364492829701001689555, 4.78853758723080216506020188348, 5.01934241822842419254120534218, 5.44836009546409179146669427874, 5.83076933706513894768780898637, 6.47508992198743382441741548276, 7.35894510818848211701162857017, 7.36332304463235693496451944900, 7.79255165723859507121450273367, 7.966098742367669350050663928217, 8.324780208928235940542224213421, 8.753458223904739243849774299404, 9.506858341875896690407733367347, 9.523016099171869295072700801870
