| L(s) = 1 | − 4.01e3·13-s + 1.56e5·25-s + 6.71e5·37-s − 7.20e4·49-s + 3.99e6·61-s − 1.00e7·73-s + 3.50e7·97-s + 3.36e7·109-s − 3.89e7·121-s + ⋯ |
| L(s) = 1 | − 0.507·13-s + 2·25-s + 2.17·37-s − 0.0875·49-s + 2.25·61-s − 3.03·73-s + 3.89·97-s + 2.48·109-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.404589317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.404589317\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 1255 T + p^{7} T^{2} )( 1 + 1255 T + p^{7} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2009 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 43091 T + p^{7} T^{2} )( 1 + 43091 T + p^{7} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 178916 T + p^{7} T^{2} )( 1 + 178916 T + p^{7} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 335663 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1035224 T + p^{7} T^{2} )( 1 + 1035224 T + p^{7} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1998347 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4443527 T + p^{7} T^{2} )( 1 + 4443527 T + p^{7} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5038001 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4517617 T + p^{7} T^{2} )( 1 + 4517617 T + p^{7} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17521555 T + p^{7} T^{2} )^{2} \) |
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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
−10.22049484307329948857503802086, −9.883303599048793718670554128933, −9.114568968763766871395200819494, −9.013443636321387244724699439862, −8.419642643509299366262478892497, −7.948080139671281530909254122082, −7.29004649014500331654892520109, −7.20129487244212871346308501082, −6.30553673803786016630166747900, −6.23488333502209820308880877657, −5.37163478861667416973224450236, −5.02851495571833788372680179377, −4.41067664355203410532110305506, −4.07103165710533867103889013052, −3.13808331732563985853069196018, −2.89133679367636399211919381724, −2.22399930758774055192879402175, −1.59832174099828690472475507208, −0.69444590969990739167135937345, −0.63850782651936981137134717327,
0.63850782651936981137134717327, 0.69444590969990739167135937345, 1.59832174099828690472475507208, 2.22399930758774055192879402175, 2.89133679367636399211919381724, 3.13808331732563985853069196018, 4.07103165710533867103889013052, 4.41067664355203410532110305506, 5.02851495571833788372680179377, 5.37163478861667416973224450236, 6.23488333502209820308880877657, 6.30553673803786016630166747900, 7.20129487244212871346308501082, 7.29004649014500331654892520109, 7.948080139671281530909254122082, 8.419642643509299366262478892497, 9.013443636321387244724699439862, 9.114568968763766871395200819494, 9.883303599048793718670554128933, 10.22049484307329948857503802086
