L(s) = 1 | + 4-s − 3·9-s + 4·13-s + 16-s + 8·19-s − 6·25-s − 2·31-s − 3·36-s + 20·37-s + 16·43-s − 14·49-s + 4·52-s − 12·61-s + 64-s − 24·67-s + 20·73-s + 8·76-s − 16·79-s + 9·81-s + 4·97-s − 6·100-s + 16·103-s − 4·109-s − 12·117-s − 22·121-s − 2·124-s + 127-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 9-s + 1.10·13-s + 1/4·16-s + 1.83·19-s − 6/5·25-s − 0.359·31-s − 1/2·36-s + 3.28·37-s + 2.43·43-s − 2·49-s + 0.554·52-s − 1.53·61-s + 1/8·64-s − 2.93·67-s + 2.34·73-s + 0.917·76-s − 1.80·79-s + 81-s + 0.406·97-s − 3/5·100-s + 1.57·103-s − 0.383·109-s − 1.10·117-s − 2·121-s − 0.179·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34596 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.433324264\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.433324264\) |
\(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.50978028730293557207764308625, −9.792819761543788827804434951174, −9.258085561472781634099322215629, −9.056498267458590117851162621111, −8.059614510201906936269350354648, −7.76857292337562945056886541715, −7.42375835124516750954422223560, −6.32858579420662158347624019611, −6.01709210768183680394437750392, −5.66012783592677814232561850819, −4.74203912193695697715499891246, −3.93624561426330149363253797109, −3.16188554299090604324575557253, −2.56538619972077555026069053800, −1.25027517360414188495745145887,
1.25027517360414188495745145887, 2.56538619972077555026069053800, 3.16188554299090604324575557253, 3.93624561426330149363253797109, 4.74203912193695697715499891246, 5.66012783592677814232561850819, 6.01709210768183680394437750392, 6.32858579420662158347624019611, 7.42375835124516750954422223560, 7.76857292337562945056886541715, 8.059614510201906936269350354648, 9.056498267458590117851162621111, 9.258085561472781634099322215629, 9.792819761543788827804434951174, 10.50978028730293557207764308625