| L(s) = 1 | − 2-s + 4-s − 8-s + 4·9-s − 8·13-s + 16-s + 3·17-s − 4·18-s − 25-s + 8·26-s + 3·29-s − 32-s − 3·34-s + 4·36-s − 14·37-s − 9·41-s − 4·49-s + 50-s − 8·52-s − 3·53-s − 3·58-s + 7·61-s + 64-s + 3·68-s − 4·72-s + 13·73-s + 14·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 4/3·9-s − 2.21·13-s + 1/4·16-s + 0.727·17-s − 0.942·18-s − 1/5·25-s + 1.56·26-s + 0.557·29-s − 0.176·32-s − 0.514·34-s + 2/3·36-s − 2.30·37-s − 1.40·41-s − 4/7·49-s + 0.141·50-s − 1.10·52-s − 0.412·53-s − 0.393·58-s + 0.896·61-s + 1/8·64-s + 0.363·68-s − 0.471·72-s + 1.52·73-s + 1.62·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5630230100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5630230100\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 10 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 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
−12.59513878661946422239236506661, −12.10073323528103126700108361109, −11.82107789688609435951988745699, −10.71646527624477058060356596209, −10.15612587466812377185969794871, −9.873821433732743264824284214941, −9.287390939168912817848315082256, −8.365129497891204962528650330030, −7.67725494470328296816291198344, −7.09254118450880494451081790652, −6.65142011489218057679269267791, −5.30571758222403800950190730640, −4.71687777602847961177846742985, −3.41555289929474868212653451403, −2.00085898254727558063116708811,
2.00085898254727558063116708811, 3.41555289929474868212653451403, 4.71687777602847961177846742985, 5.30571758222403800950190730640, 6.65142011489218057679269267791, 7.09254118450880494451081790652, 7.67725494470328296816291198344, 8.365129497891204962528650330030, 9.287390939168912817848315082256, 9.873821433732743264824284214941, 10.15612587466812377185969794871, 10.71646527624477058060356596209, 11.82107789688609435951988745699, 12.10073323528103126700108361109, 12.59513878661946422239236506661
