| L(s) = 1 | + 2·2-s + 3·4-s + 3·7-s + 4·8-s − 5·9-s + 4·11-s + 6·14-s + 5·16-s − 10·18-s + 8·22-s − 12·23-s − 9·25-s + 9·28-s − 20·29-s + 6·32-s − 15·36-s − 4·37-s + 8·43-s + 12·44-s − 24·46-s + 2·49-s − 18·50-s + 8·53-s + 12·56-s − 40·58-s − 15·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.13·7-s + 1.41·8-s − 5/3·9-s + 1.20·11-s + 1.60·14-s + 5/4·16-s − 2.35·18-s + 1.70·22-s − 2.50·23-s − 9/5·25-s + 1.70·28-s − 3.71·29-s + 1.06·32-s − 5/2·36-s − 0.657·37-s + 1.21·43-s + 1.80·44-s − 3.53·46-s + 2/7·49-s − 2.54·50-s + 1.09·53-s + 1.60·56-s − 5.25·58-s − 1.88·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1223236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1223236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 79 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−7.74010287138397763242022016623, −7.50453196645377754362608030446, −6.79143285188373633784741807428, −6.11586748167354289065343430906, −5.89722887687190693675870945379, −5.60309356526707119157118370453, −5.34006059048988750874758197539, −4.49924691164990011239017251780, −4.08090613362146692342747948187, −3.69696678277400607803844195002, −3.45126436232312497697169849903, −2.34624317045513577735973692504, −2.06983106012671431074676520675, −1.59907559861217528540567634463, 0,
1.59907559861217528540567634463, 2.06983106012671431074676520675, 2.34624317045513577735973692504, 3.45126436232312497697169849903, 3.69696678277400607803844195002, 4.08090613362146692342747948187, 4.49924691164990011239017251780, 5.34006059048988750874758197539, 5.60309356526707119157118370453, 5.89722887687190693675870945379, 6.11586748167354289065343430906, 6.79143285188373633784741807428, 7.50453196645377754362608030446, 7.74010287138397763242022016623
