| L(s) = 1 | + 2·7-s − 20·31-s − 22·37-s − 2·43-s + 3·49-s − 16·61-s + 6·67-s + 20·73-s − 22·79-s − 4·97-s + 16·103-s + 6·109-s − 15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 3.59·31-s − 3.61·37-s − 0.304·43-s + 3/7·49-s − 2.04·61-s + 0.733·67-s + 2.34·73-s − 2.47·79-s − 0.406·97-s + 1.57·103-s + 0.574·109-s − 1.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 135 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 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
−7.78049290473661850625037622372, −7.47296742311939075860670516664, −7.12080713195526744616103154657, −7.06069018890767508642474962665, −6.39728194182531935636266108157, −6.16119604685960460710550776666, −5.51788390511546958224210991533, −5.40061961017984438594880110464, −4.92133844422250002304179349013, −4.91748212917055359219610410262, −3.97733507807368808124983850106, −3.93378091201686525303816031148, −3.35886635460672454845798409320, −3.23460373625321879357961837379, −2.27842754128221222166853911887, −2.14667339897850451554152562933, −1.45046505865770407585813835701, −1.37348103522673980974239079577, 0, 0,
1.37348103522673980974239079577, 1.45046505865770407585813835701, 2.14667339897850451554152562933, 2.27842754128221222166853911887, 3.23460373625321879357961837379, 3.35886635460672454845798409320, 3.93378091201686525303816031148, 3.97733507807368808124983850106, 4.91748212917055359219610410262, 4.92133844422250002304179349013, 5.40061961017984438594880110464, 5.51788390511546958224210991533, 6.16119604685960460710550776666, 6.39728194182531935636266108157, 7.06069018890767508642474962665, 7.12080713195526744616103154657, 7.47296742311939075860670516664, 7.78049290473661850625037622372
