| L(s) = 1 | + 2·9-s − 4·13-s + 3·17-s − 6·25-s − 8·29-s − 4·41-s + 2·49-s + 4·53-s − 4·73-s − 5·81-s + 20·89-s + 20·97-s − 4·101-s + 4·113-s − 8·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 6·153-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 1.10·13-s + 0.727·17-s − 6/5·25-s − 1.48·29-s − 0.624·41-s + 2/7·49-s + 0.549·53-s − 0.468·73-s − 5/9·81-s + 2.11·89-s + 2.03·97-s − 0.398·101-s + 0.376·113-s − 0.739·117-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.485·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8081985183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8081985183\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−12.30387478799107737983373508846, −11.90214162552801180036419415071, −11.34828654055755564419541198678, −10.47937801451071443935866756702, −10.03798284985685875805533854093, −9.511142428036285296638299086377, −8.905038179637674254269327250075, −7.895547330772521781176071648221, −7.52318353687672249642423356056, −6.87573751028600003994577244954, −5.90643760125537322887459729980, −5.23301239029312200306377698753, −4.33139004401435712729046835691, −3.42920770610915173809752336250, −2.04142023158273960704107134446,
2.04142023158273960704107134446, 3.42920770610915173809752336250, 4.33139004401435712729046835691, 5.23301239029312200306377698753, 5.90643760125537322887459729980, 6.87573751028600003994577244954, 7.52318353687672249642423356056, 7.895547330772521781176071648221, 8.905038179637674254269327250075, 9.511142428036285296638299086377, 10.03798284985685875805533854093, 10.47937801451071443935866756702, 11.34828654055755564419541198678, 11.90214162552801180036419415071, 12.30387478799107737983373508846
