| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·8-s − 10-s + 11-s − 12-s + 13-s + 15-s + 2·16-s + 2·19-s − 20-s + 22-s − 2·24-s + 26-s + 27-s + 30-s + 2·32-s − 33-s − 2·37-s + 2·38-s − 39-s − 2·40-s + 44-s − 2·48-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·8-s − 10-s + 11-s − 12-s + 13-s + 15-s + 2·16-s + 2·19-s − 20-s + 22-s − 2·24-s + 26-s + 27-s + 30-s + 2·32-s − 33-s − 2·37-s + 2·38-s − 39-s − 2·40-s + 44-s − 2·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.321877503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321877503\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−10.67902765044076885685885639404, −10.62263112617358495073049964707, −9.800251267550109782204396607383, −9.499499425048540020274206872204, −8.871761633891009084291801383350, −8.273782596628958866388757061487, −7.920804388662517541291852012680, −7.52378973501114074844991737809, −7.01066731434009246166094142285, −6.69317900147863393963269333666, −6.25648599734183195164879183440, −5.53196818025767612196188082667, −5.49174128296106319361309973377, −4.67521868356753176431705395629, −4.47188274104729856876338173000, −3.87517981966162602449470147794, −3.22144856179938156064369391772, −3.11011177603211712190796048487, −1.58946467751898220065287823715, −1.35611621016606595567620269397,
1.35611621016606595567620269397, 1.58946467751898220065287823715, 3.11011177603211712190796048487, 3.22144856179938156064369391772, 3.87517981966162602449470147794, 4.47188274104729856876338173000, 4.67521868356753176431705395629, 5.49174128296106319361309973377, 5.53196818025767612196188082667, 6.25648599734183195164879183440, 6.69317900147863393963269333666, 7.01066731434009246166094142285, 7.52378973501114074844991737809, 7.920804388662517541291852012680, 8.273782596628958866388757061487, 8.871761633891009084291801383350, 9.499499425048540020274206872204, 9.800251267550109782204396607383, 10.62263112617358495073049964707, 10.67902765044076885685885639404
