| L(s) = 1 | + 3·4-s − 5-s − 4·9-s + 6·11-s + 5·16-s − 2·19-s − 3·20-s + 25-s − 6·31-s − 12·36-s + 18·44-s + 4·45-s − 6·49-s − 6·55-s + 6·59-s + 4·61-s + 3·64-s − 12·71-s − 6·76-s − 10·79-s − 5·80-s + 7·81-s + 2·95-s − 24·99-s + 3·100-s + 6·101-s − 34·109-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.447·5-s − 4/3·9-s + 1.80·11-s + 5/4·16-s − 0.458·19-s − 0.670·20-s + 1/5·25-s − 1.07·31-s − 2·36-s + 2.71·44-s + 0.596·45-s − 6/7·49-s − 0.809·55-s + 0.781·59-s + 0.512·61-s + 3/8·64-s − 1.42·71-s − 0.688·76-s − 1.12·79-s − 0.559·80-s + 7/9·81-s + 0.205·95-s − 2.41·99-s + 3/10·100-s + 0.597·101-s − 3.25·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358773366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358773366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
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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
−11.11015621985125861482579848476, −10.97682742951522138975397195369, −10.14778824531746060665288486773, −9.430399448324839804875505020280, −8.823493013273446188590564152732, −8.412737890333081510294586277526, −7.65183287082923003663822040057, −7.02268288271653419256557945483, −6.57287630160273962471142035588, −6.03502205951233022243340305084, −5.42515198321540359629076991699, −4.26849477194728750023117848663, −3.50637573664963150571858013304, −2.74932061012723778852372669392, −1.68835472074481476208269751200,
1.68835472074481476208269751200, 2.74932061012723778852372669392, 3.50637573664963150571858013304, 4.26849477194728750023117848663, 5.42515198321540359629076991699, 6.03502205951233022243340305084, 6.57287630160273962471142035588, 7.02268288271653419256557945483, 7.65183287082923003663822040057, 8.412737890333081510294586277526, 8.823493013273446188590564152732, 9.430399448324839804875505020280, 10.14778824531746060665288486773, 10.97682742951522138975397195369, 11.11015621985125861482579848476
