| L(s) = 1 | − 2·5-s + 2·7-s + 11-s + 13-s − 11·17-s − 6·19-s + 2·23-s + 3·25-s − 5·29-s − 4·31-s − 4·35-s − 6·41-s − 6·43-s − 9·47-s + 3·49-s + 18·53-s − 2·55-s + 8·59-s − 22·61-s − 2·65-s − 12·67-s + 8·73-s + 2·77-s − 11·79-s − 24·83-s + 22·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 0.301·11-s + 0.277·13-s − 2.66·17-s − 1.37·19-s + 0.417·23-s + 3/5·25-s − 0.928·29-s − 0.718·31-s − 0.676·35-s − 0.937·41-s − 0.914·43-s − 1.31·47-s + 3/7·49-s + 2.47·53-s − 0.269·55-s + 1.04·59-s − 2.81·61-s − 0.248·65-s − 1.46·67-s + 0.936·73-s + 0.227·77-s − 1.23·79-s − 2.63·83-s + 2.38·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 110 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 22 T + 226 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 150 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 212 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.657421573595445861698193807345, −8.542005645504926115697183624833, −8.036228836198284947919140686695, −7.46340251263528762901116067456, −7.29996443665443822734211661285, −6.78129836074466195756697846011, −6.41004369607984789563315397320, −6.28538090646005363654844159013, −5.38847663362943702159840931612, −5.21521660863158836028815157688, −4.60330385739275168207884621491, −4.32030328427404534032647059087, −3.89902830612069211860597205348, −3.70963510305561155013139119065, −2.65771925232880465484349677960, −2.60504766567610760160526252547, −1.59201670898026079357220382839, −1.57644727734997939278034897743, 0, 0,
1.57644727734997939278034897743, 1.59201670898026079357220382839, 2.60504766567610760160526252547, 2.65771925232880465484349677960, 3.70963510305561155013139119065, 3.89902830612069211860597205348, 4.32030328427404534032647059087, 4.60330385739275168207884621491, 5.21521660863158836028815157688, 5.38847663362943702159840931612, 6.28538090646005363654844159013, 6.41004369607984789563315397320, 6.78129836074466195756697846011, 7.29996443665443822734211661285, 7.46340251263528762901116067456, 8.036228836198284947919140686695, 8.542005645504926115697183624833, 8.657421573595445861698193807345
