| L(s) = 1 | − 3-s − 19-s + 25-s + 27-s + 31-s − 37-s + 57-s − 3·67-s + 3·73-s − 75-s + 3·79-s − 81-s − 93-s + 103-s + 109-s + 111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
| L(s) = 1 | − 3-s − 19-s + 25-s + 27-s + 31-s − 37-s + 57-s − 3·67-s + 3·73-s − 75-s + 3·79-s − 81-s − 93-s + 103-s + 109-s + 111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7312766283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7312766283\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.208402561784249129403354128363, −9.027932805887989842587486150410, −8.507345367718888280854755979823, −8.387974152592066553983314740843, −7.68773764055467617950474497285, −7.54690697282397223345786221013, −6.83030911364757468548076554435, −6.54970445693668472833477945798, −6.35261099496016856862426474389, −5.91883268400493973093050510489, −5.37539195037049026172575645262, −5.07767896348740183805831303840, −4.58410708430356239827825179234, −4.42305005267351053565529601173, −3.51833932078734243804777823229, −3.38146687422328634116103421430, −2.57378002851777410738913291666, −2.20767064479174365444230827841, −1.41000418042886420178066325177, −0.65229264616749066761673331274,
0.65229264616749066761673331274, 1.41000418042886420178066325177, 2.20767064479174365444230827841, 2.57378002851777410738913291666, 3.38146687422328634116103421430, 3.51833932078734243804777823229, 4.42305005267351053565529601173, 4.58410708430356239827825179234, 5.07767896348740183805831303840, 5.37539195037049026172575645262, 5.91883268400493973093050510489, 6.35261099496016856862426474389, 6.54970445693668472833477945798, 6.83030911364757468548076554435, 7.54690697282397223345786221013, 7.68773764055467617950474497285, 8.387974152592066553983314740843, 8.507345367718888280854755979823, 9.027932805887989842587486150410, 9.208402561784249129403354128363
