| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 5·9-s − 6·11-s − 4·14-s + 5·16-s + 10·18-s + 12·22-s + 12·23-s + 6·28-s − 6·32-s − 15·36-s + 4·37-s − 8·43-s − 18·44-s − 24·46-s − 3·49-s + 12·53-s − 8·56-s − 10·63-s + 7·64-s − 26·67-s + 24·71-s + 20·72-s − 8·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 5/3·9-s − 1.80·11-s − 1.06·14-s + 5/4·16-s + 2.35·18-s + 2.55·22-s + 2.50·23-s + 1.13·28-s − 1.06·32-s − 5/2·36-s + 0.657·37-s − 1.21·43-s − 2.71·44-s − 3.53·46-s − 3/7·49-s + 1.64·53-s − 1.06·56-s − 1.25·63-s + 7/8·64-s − 3.17·67-s + 2.84·71-s + 2.35·72-s − 0.929·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.194307401639253088525291180127, −8.521585518702256458221706662325, −8.316812877075225116243141130520, −7.993773173422627541887367381542, −7.29853134793764088505009214313, −7.02301401808588593198377416434, −6.22236976774475581546663735816, −5.61328186020314811053228254285, −5.20592148788576143134616479765, −4.75932804332696589670523572045, −3.45948532224630523972346946948, −2.65155040417235216240951299295, −2.60475259048520463737512973942, −1.28122616800987786392826491244, 0,
1.28122616800987786392826491244, 2.60475259048520463737512973942, 2.65155040417235216240951299295, 3.45948532224630523972346946948, 4.75932804332696589670523572045, 5.20592148788576143134616479765, 5.61328186020314811053228254285, 6.22236976774475581546663735816, 7.02301401808588593198377416434, 7.29853134793764088505009214313, 7.993773173422627541887367381542, 8.316812877075225116243141130520, 8.521585518702256458221706662325, 9.194307401639253088525291180127
