| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 2·9-s + 2·14-s + 16-s − 8·17-s + 2·18-s + 6·25-s + 2·28-s − 8·31-s + 32-s − 8·34-s + 2·36-s + 2·41-s − 2·49-s + 6·50-s + 2·56-s − 8·62-s + 4·63-s + 64-s − 8·68-s − 26·71-s + 2·72-s + 10·73-s + 12·79-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 2/3·9-s + 0.534·14-s + 1/4·16-s − 1.94·17-s + 0.471·18-s + 6/5·25-s + 0.377·28-s − 1.43·31-s + 0.176·32-s − 1.37·34-s + 1/3·36-s + 0.312·41-s − 2/7·49-s + 0.848·50-s + 0.267·56-s − 1.01·62-s + 0.503·63-s + 1/8·64-s − 0.970·68-s − 3.08·71-s + 0.235·72-s + 1.17·73-s + 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.954577372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954577372\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 191 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.95008120266292873022175901808, −10.44148275679474565556533429165, −9.586407670474221026333488452660, −9.061528948735591450622931255273, −8.584795753660167349179589489214, −7.899777852963101087055247769971, −7.23351486803106688335957354101, −6.85168764726258654568996526926, −6.22758167001922351138783556937, −5.44620695850531294568913618499, −4.72955057373559863679461202346, −4.40816482597612371775129873399, −3.57841859675043089494346383461, −2.52305410225181858305777558421, −1.65759590861630146172915238145,
1.65759590861630146172915238145, 2.52305410225181858305777558421, 3.57841859675043089494346383461, 4.40816482597612371775129873399, 4.72955057373559863679461202346, 5.44620695850531294568913618499, 6.22758167001922351138783556937, 6.85168764726258654568996526926, 7.23351486803106688335957354101, 7.899777852963101087055247769971, 8.584795753660167349179589489214, 9.061528948735591450622931255273, 9.586407670474221026333488452660, 10.44148275679474565556533429165, 10.95008120266292873022175901808
