L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 8·7-s − 8-s + 9-s − 10-s + 12-s − 4·13-s − 8·14-s + 15-s + 16-s + 12·17-s − 18-s − 8·19-s + 20-s + 8·21-s − 24-s + 25-s + 4·26-s + 27-s + 8·28-s + 12·29-s − 30-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 3.02·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.10·13-s − 2.13·14-s + 0.258·15-s + 1/4·16-s + 2.91·17-s − 0.235·18-s − 1.83·19-s + 0.223·20-s + 1.74·21-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 1.51·28-s + 2.22·29-s − 0.182·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.939894875\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.939894875\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 - T \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.307525687782695431948900879973, −8.128521468164053694071078641328, −7.84831740970949093070141074865, −7.71052407307967077644420749342, −6.76895238282715006434981339956, −6.58767052470959898247985953468, −5.48025531888955534300415720953, −5.37349128799592068356886197367, −4.84143328212506167706839183620, −4.36655234059998826349058539516, −3.69936628544405197022333195268, −2.65454187476862090428654397793, −2.39041205999551187286748903960, −1.43294542786364492056174526795, −1.28283795797463795719739258116,
1.28283795797463795719739258116, 1.43294542786364492056174526795, 2.39041205999551187286748903960, 2.65454187476862090428654397793, 3.69936628544405197022333195268, 4.36655234059998826349058539516, 4.84143328212506167706839183620, 5.37349128799592068356886197367, 5.48025531888955534300415720953, 6.58767052470959898247985953468, 6.76895238282715006434981339956, 7.71052407307967077644420749342, 7.84831740970949093070141074865, 8.128521468164053694071078641328, 8.307525687782695431948900879973