| L(s) = 1 | + 4-s − 4·9-s − 6·11-s + 16-s + 10·23-s − 25-s + 6·29-s − 4·36-s + 4·37-s − 6·43-s − 6·44-s − 14·53-s + 64-s − 16·67-s − 10·71-s − 2·79-s + 7·81-s + 10·92-s + 24·99-s − 100-s − 26·109-s − 16·113-s + 6·116-s + 6·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 4/3·9-s − 1.80·11-s + 1/4·16-s + 2.08·23-s − 1/5·25-s + 1.11·29-s − 2/3·36-s + 0.657·37-s − 0.914·43-s − 0.904·44-s − 1.92·53-s + 1/8·64-s − 1.95·67-s − 1.18·71-s − 0.225·79-s + 7/9·81-s + 1.04·92-s + 2.41·99-s − 0.0999·100-s − 2.49·109-s − 1.50·113-s + 0.557·116-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 100 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 162 T^{2} + p^{2} T^{4} \) |
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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
−8.685196910846543852882243263747, −8.215792129540017999929692237521, −7.79782276235516778954655126577, −7.47713827367886104326913061119, −6.73260846270624383241870044269, −6.38702790674810038003241934846, −5.77445857996051476792035156015, −5.22446408057072142587745122266, −5.00416527922499528519906763070, −4.28559822650419719186187543650, −3.18130525758676255285434406358, −2.89503454526385376089488787089, −2.56012496327512619236083548923, −1.39029805183652987640412620385, 0,
1.39029805183652987640412620385, 2.56012496327512619236083548923, 2.89503454526385376089488787089, 3.18130525758676255285434406358, 4.28559822650419719186187543650, 5.00416527922499528519906763070, 5.22446408057072142587745122266, 5.77445857996051476792035156015, 6.38702790674810038003241934846, 6.73260846270624383241870044269, 7.47713827367886104326913061119, 7.79782276235516778954655126577, 8.215792129540017999929692237521, 8.685196910846543852882243263747
