| L(s) = 1 | − 4-s + 4·7-s − 2·8-s − 4·9-s + 4·11-s + 4·13-s + 16-s − 6·25-s − 4·28-s + 12·31-s + 4·32-s + 4·36-s + 8·41-s − 4·44-s + 2·49-s − 4·52-s + 4·53-s − 8·56-s − 12·61-s − 16·63-s + 3·64-s + 8·72-s + 16·77-s + 7·81-s + 12·83-s − 8·88-s + 16·91-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.51·7-s − 0.707·8-s − 4/3·9-s + 1.20·11-s + 1.10·13-s + 1/4·16-s − 6/5·25-s − 0.755·28-s + 2.15·31-s + 0.707·32-s + 2/3·36-s + 1.24·41-s − 0.603·44-s + 2/7·49-s − 0.554·52-s + 0.549·53-s − 1.06·56-s − 1.53·61-s − 2.01·63-s + 3/8·64-s + 0.942·72-s + 1.82·77-s + 7/9·81-s + 1.31·83-s − 0.852·88-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25538 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25538 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.195737336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.195737336\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 113 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 100 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.99339513567532214010358711057, −10.03738908888083438005343610998, −9.482739262741147690691968106192, −8.920098062622682066518846558395, −8.522612671281386109551214519313, −8.143959498729516225099390774301, −7.66564172593200134495041605676, −6.55365565146131181831641349725, −6.09300940768430969280589724005, −5.67610659684221795644477219111, −4.82704049962210373706393517645, −4.22376300887116086081841179410, −3.49656800984114260862676677094, −2.53977820924588088282769973173, −1.25138314974718007440597741029,
1.25138314974718007440597741029, 2.53977820924588088282769973173, 3.49656800984114260862676677094, 4.22376300887116086081841179410, 4.82704049962210373706393517645, 5.67610659684221795644477219111, 6.09300940768430969280589724005, 6.55365565146131181831641349725, 7.66564172593200134495041605676, 8.143959498729516225099390774301, 8.522612671281386109551214519313, 8.920098062622682066518846558395, 9.482739262741147690691968106192, 10.03738908888083438005343610998, 10.99339513567532214010358711057
