| L(s) = 1 | + 2·13-s + 25-s + 4·37-s − 49-s − 2·61-s − 4·73-s + 2·97-s − 4·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2·13-s + 25-s + 4·37-s − 49-s − 2·61-s − 4·73-s + 2·97-s − 4·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.247033888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.247033888\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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
−9.826207567758078568599489847543, −9.799943370483185191917632267189, −9.024673114278215893458209243182, −8.998251954515816010525002972349, −8.436774389104417240398597665973, −8.089965652763877437246867911823, −7.56665921137478423816141511945, −7.40164564533685454789307351100, −6.53338037649943581419368298978, −6.35948809764005141177484987876, −5.92012251779772620727959588417, −5.70909325345432036401935897155, −4.76514537377726367324841783769, −4.60813037763135039422912437780, −3.99686914297187884985064830764, −3.59016694608725740294725653465, −2.77700903918408898284677472720, −2.71058323338231288014984230798, −1.46977545229909853101001693791, −1.16768116849375470807889173288,
1.16768116849375470807889173288, 1.46977545229909853101001693791, 2.71058323338231288014984230798, 2.77700903918408898284677472720, 3.59016694608725740294725653465, 3.99686914297187884985064830764, 4.60813037763135039422912437780, 4.76514537377726367324841783769, 5.70909325345432036401935897155, 5.92012251779772620727959588417, 6.35948809764005141177484987876, 6.53338037649943581419368298978, 7.40164564533685454789307351100, 7.56665921137478423816141511945, 8.089965652763877437246867911823, 8.436774389104417240398597665973, 8.998251954515816010525002972349, 9.024673114278215893458209243182, 9.799943370483185191917632267189, 9.826207567758078568599489847543
