| L(s) = 1 | + 7-s − 2·9-s + 8·11-s − 8·23-s + 6·25-s − 8·29-s + 8·37-s + 8·43-s + 49-s + 16·53-s − 2·63-s − 16·71-s + 8·77-s − 5·81-s − 16·99-s + 24·109-s + 12·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 8·161-s + 163-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 2/3·9-s + 2.41·11-s − 1.66·23-s + 6/5·25-s − 1.48·29-s + 1.31·37-s + 1.21·43-s + 1/7·49-s + 2.19·53-s − 0.251·63-s − 1.89·71-s + 0.911·77-s − 5/9·81-s − 1.60·99-s + 2.29·109-s + 1.12·113-s + 2.36·121-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 + 0.0798·157-s − 0.630·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091952824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091952824\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−8.872579179288820817899580669762, −8.394490584499298365663162677709, −7.81987907595241636425503651280, −7.20983952030760728435052345400, −7.00968721240317035620303706007, −6.24986029304499473922534359064, −5.87684815622633737979183709140, −5.67671568401614686099789109093, −4.69505800909619056415588298667, −4.22074959501737259101638295571, −3.88633796042519497912112110541, −3.24919851939206442448002093540, −2.39541894189509685594019786714, −1.72159178855419828072310958108, −0.873134050861652498194030934706,
0.873134050861652498194030934706, 1.72159178855419828072310958108, 2.39541894189509685594019786714, 3.24919851939206442448002093540, 3.88633796042519497912112110541, 4.22074959501737259101638295571, 4.69505800909619056415588298667, 5.67671568401614686099789109093, 5.87684815622633737979183709140, 6.24986029304499473922534359064, 7.00968721240317035620303706007, 7.20983952030760728435052345400, 7.81987907595241636425503651280, 8.394490584499298365663162677709, 8.872579179288820817899580669762
