| L(s) = 1 | + 2·3-s − 4-s − 2·7-s − 6·11-s − 2·12-s − 3·16-s − 12·17-s − 4·19-s − 4·21-s + 6·23-s − 7·25-s − 2·27-s + 2·28-s − 6·29-s + 2·31-s − 12·33-s + 14·37-s + 10·43-s + 6·44-s − 12·47-s − 6·48-s + 3·49-s − 24·51-s − 6·53-s − 8·57-s − 18·59-s − 20·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 0.755·7-s − 1.80·11-s − 0.577·12-s − 3/4·16-s − 2.91·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s − 7/5·25-s − 0.384·27-s + 0.377·28-s − 1.11·29-s + 0.359·31-s − 2.08·33-s + 2.30·37-s + 1.52·43-s + 0.904·44-s − 1.75·47-s − 0.866·48-s + 3/7·49-s − 3.36·51-s − 0.824·53-s − 1.05·57-s − 2.34·59-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 195 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 108 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 123 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.360438878349297248235224639730, −9.318431514677108533108845399192, −8.649628853872145341367351689353, −8.554796684694620869248593087602, −7.983608496186141154462739630124, −7.51821351177049365235861067884, −7.38059187606478749895460845120, −6.52901228257874317659307463150, −6.18826341062502788637453866112, −6.01847552641775700461429554518, −5.08770078656462847327918556255, −4.68550777932376371925645836081, −4.36832608437226875511155361649, −3.91086875357420949604621609186, −2.99324455808367928275809900809, −2.79950106455141729889317959166, −2.37190935564413613059173871538, −1.82481648344398235946456611792, 0, 0,
1.82481648344398235946456611792, 2.37190935564413613059173871538, 2.79950106455141729889317959166, 2.99324455808367928275809900809, 3.91086875357420949604621609186, 4.36832608437226875511155361649, 4.68550777932376371925645836081, 5.08770078656462847327918556255, 6.01847552641775700461429554518, 6.18826341062502788637453866112, 6.52901228257874317659307463150, 7.38059187606478749895460845120, 7.51821351177049365235861067884, 7.983608496186141154462739630124, 8.554796684694620869248593087602, 8.649628853872145341367351689353, 9.318431514677108533108845399192, 9.360438878349297248235224639730
