| L(s) = 1 | + 7-s + 2·9-s − 4·11-s + 4·23-s + 6·25-s + 16·29-s + 8·37-s + 12·43-s + 49-s + 4·53-s + 2·63-s − 4·77-s − 5·81-s − 8·99-s − 32·107-s + 16·109-s + 4·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + 163-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 2/3·9-s − 1.20·11-s + 0.834·23-s + 6/5·25-s + 2.97·29-s + 1.31·37-s + 1.82·43-s + 1/7·49-s + 0.549·53-s + 0.251·63-s − 0.455·77-s − 5/9·81-s − 0.804·99-s − 3.09·107-s + 1.53·109-s + 0.376·113-s − 0.545·121-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 + 0.315·161-s + 0.0783·163-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
| \(L(1)\) |
\(\approx\) |
\(2.624694380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.624694380\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 170 T^{2} + p^{2} T^{4} \) |
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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
−7.88141621718185587197034632050, −7.58252574174644448721066256413, −7.06041050174913625098611822083, −6.72004791321776142968750236016, −6.27369448907204373654297099637, −5.69483165949111729900547756684, −5.24699196417129412925415384478, −4.70557549370005929953803756175, −4.52888646732450766791406891014, −3.98804946472276595480203988764, −3.10928311745819888460525767719, −2.69646086029005397649121999911, −2.38682896528902303409865720605, −1.25985139020924222725570342395, −0.810851731601587938385971351199,
0.810851731601587938385971351199, 1.25985139020924222725570342395, 2.38682896528902303409865720605, 2.69646086029005397649121999911, 3.10928311745819888460525767719, 3.98804946472276595480203988764, 4.52888646732450766791406891014, 4.70557549370005929953803756175, 5.24699196417129412925415384478, 5.69483165949111729900547756684, 6.27369448907204373654297099637, 6.72004791321776142968750236016, 7.06041050174913625098611822083, 7.58252574174644448721066256413, 7.88141621718185587197034632050
