| L(s) = 1 | + 3-s + 7-s − 4·9-s − 11-s + 3·13-s − 17-s + 3·19-s + 21-s + 2·23-s − 6·27-s − 14·29-s − 7·31-s − 33-s − 4·37-s + 3·39-s − 9·41-s + 6·47-s − 12·49-s − 51-s + 8·53-s + 3·57-s + 10·59-s − 3·61-s − 4·63-s + 20·67-s + 2·69-s − 3·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 4/3·9-s − 0.301·11-s + 0.832·13-s − 0.242·17-s + 0.688·19-s + 0.218·21-s + 0.417·23-s − 1.15·27-s − 2.59·29-s − 1.25·31-s − 0.174·33-s − 0.657·37-s + 0.480·39-s − 1.40·41-s + 0.875·47-s − 1.71·49-s − 0.140·51-s + 1.09·53-s + 0.397·57-s + 1.30·59-s − 0.384·61-s − 0.503·63-s + 2.44·67-s + 0.240·69-s − 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 27 T + 375 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
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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
−7.44327817957837560709447162499, −7.40177112914968443631161622888, −6.83263844198356518443865782104, −6.69397454932319903239494361539, −6.10984777838260858791294886957, −5.63088064846037893097079026398, −5.43858820748899125312616633818, −5.37714589210250908383972998923, −4.93502472001791574286311181138, −4.26413467308948680632130394897, −3.79519903072631243686626621182, −3.70079567952385446561710356643, −3.25523281088490358691779076833, −2.91518896728280599014475693028, −2.33875503361281499531163668530, −2.08697807317832002030167907154, −1.54015812469677655808674700476, −1.11137990357888653852413018034, 0, 0,
1.11137990357888653852413018034, 1.54015812469677655808674700476, 2.08697807317832002030167907154, 2.33875503361281499531163668530, 2.91518896728280599014475693028, 3.25523281088490358691779076833, 3.70079567952385446561710356643, 3.79519903072631243686626621182, 4.26413467308948680632130394897, 4.93502472001791574286311181138, 5.37714589210250908383972998923, 5.43858820748899125312616633818, 5.63088064846037893097079026398, 6.10984777838260858791294886957, 6.69397454932319903239494361539, 6.83263844198356518443865782104, 7.40177112914968443631161622888, 7.44327817957837560709447162499
