| L(s) = 1 | + 3·3-s − 44·13-s − 26·19-s − 25·25-s − 27·27-s + 46·31-s − 26·37-s − 132·39-s − 44·43-s − 78·57-s − 74·61-s − 122·67-s + 46·73-s − 75·75-s + 142·79-s − 81·81-s + 138·93-s + 4·97-s − 194·103-s + 214·109-s − 78·111-s − 121·121-s + 127-s − 132·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3-s − 3.38·13-s − 1.36·19-s − 25-s − 27-s + 1.48·31-s − 0.702·37-s − 3.38·39-s − 1.02·43-s − 1.36·57-s − 1.21·61-s − 1.82·67-s + 0.630·73-s − 75-s + 1.79·79-s − 81-s + 1.48·93-s + 4/97·97-s − 1.88·103-s + 1.96·109-s − 0.702·111-s − 121-s + 0.00787·127-s − 1.02·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8520134616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8520134616\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 73 T + p^{2} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
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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
−10.46585338978960106948698527716, −10.17092957099179829229765038957, −9.830187807622395318128480603650, −9.251823007726348740135108796656, −9.204219592854962955375168092537, −8.318516202391227996821782381265, −8.177906159312076729411681894531, −7.58279288877201456951900338993, −7.36565784553269006973898591406, −6.70286024441232755281635274247, −6.34266505967319717829527087006, −5.52673322623685321328878916635, −5.12273017027325649984311164230, −4.38849390343872713736224695249, −4.34929001111113580725694286479, −3.20254896310358304500820519513, −2.91445358431442396938335986775, −2.09510523340834369967397111383, −2.01958346474810853171097571869, −0.29796106070280674850463531125,
0.29796106070280674850463531125, 2.01958346474810853171097571869, 2.09510523340834369967397111383, 2.91445358431442396938335986775, 3.20254896310358304500820519513, 4.34929001111113580725694286479, 4.38849390343872713736224695249, 5.12273017027325649984311164230, 5.52673322623685321328878916635, 6.34266505967319717829527087006, 6.70286024441232755281635274247, 7.36565784553269006973898591406, 7.58279288877201456951900338993, 8.177906159312076729411681894531, 8.318516202391227996821782381265, 9.204219592854962955375168092537, 9.251823007726348740135108796656, 9.830187807622395318128480603650, 10.17092957099179829229765038957, 10.46585338978960106948698527716
