| L(s) = 1 | + 7-s − 4·9-s + 2·11-s − 6·23-s + 2·25-s − 6·29-s + 2·37-s − 6·43-s + 49-s + 8·53-s − 4·63-s + 12·67-s + 12·71-s + 2·77-s + 7·81-s − 8·99-s − 12·107-s + 2·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 6·161-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 4/3·9-s + 0.603·11-s − 1.25·23-s + 2/5·25-s − 1.11·29-s + 0.328·37-s − 0.914·43-s + 1/7·49-s + 1.09·53-s − 0.503·63-s + 1.46·67-s + 1.42·71-s + 0.227·77-s + 7/9·81-s − 0.804·99-s − 1.16·107-s + 0.191·109-s − 1.63·121-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 − 0.472·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
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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
−7.928343196908504586934235161855, −7.26052178614039954076174807420, −6.85290842552243782297321324219, −6.39916730847989691864912169611, −5.93823442298871427005495687259, −5.54069845675653377448704894207, −5.18312194044676804963214360656, −4.62458187822630352965554102488, −3.93684425527240854475952388211, −3.69123231822671121566050135027, −3.05386664618913981513178273340, −2.36087838488403798550737481118, −1.96692667827429310912140277983, −1.04404691290187079653462494450, 0,
1.04404691290187079653462494450, 1.96692667827429310912140277983, 2.36087838488403798550737481118, 3.05386664618913981513178273340, 3.69123231822671121566050135027, 3.93684425527240854475952388211, 4.62458187822630352965554102488, 5.18312194044676804963214360656, 5.54069845675653377448704894207, 5.93823442298871427005495687259, 6.39916730847989691864912169611, 6.85290842552243782297321324219, 7.26052178614039954076174807420, 7.928343196908504586934235161855
