| L(s) = 1 | − 2·49-s − 2·81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 2·49-s − 2·81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8573914845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8573914845\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.13456773251351270476149503580, −6.97938978789367080321694473107, −6.73310890726855330497845231387, −6.72997129486363062025832455614, −6.54653794543390019964715516702, −5.89872368593189791540780121083, −5.85124810520019524455276968749, −5.75761131439692365322948521453, −5.65901354029478109400107757354, −5.21931043532184909409040187520, −4.91065388420061679458114897654, −4.72565034120845854573166831573, −4.34504360825903976978265621202, −4.34083696138037773506611549371, −4.26015761631709285784000364193, −3.54922042604866806431481918400, −3.33423814898099148220730399900, −3.20897824436009516815087497613, −3.18849810235471269871837163990, −2.53695735564158183465936644217, −2.23557737649768168469531998060, −2.10411430542953355455703883558, −1.53132329762609171739195829274, −1.39731271490033107163109404831, −0.69952246988121338239074732604,
0.69952246988121338239074732604, 1.39731271490033107163109404831, 1.53132329762609171739195829274, 2.10411430542953355455703883558, 2.23557737649768168469531998060, 2.53695735564158183465936644217, 3.18849810235471269871837163990, 3.20897824436009516815087497613, 3.33423814898099148220730399900, 3.54922042604866806431481918400, 4.26015761631709285784000364193, 4.34083696138037773506611549371, 4.34504360825903976978265621202, 4.72565034120845854573166831573, 4.91065388420061679458114897654, 5.21931043532184909409040187520, 5.65901354029478109400107757354, 5.75761131439692365322948521453, 5.85124810520019524455276968749, 5.89872368593189791540780121083, 6.54653794543390019964715516702, 6.72997129486363062025832455614, 6.73310890726855330497845231387, 6.97938978789367080321694473107, 7.13456773251351270476149503580
