L(s) = 1 | + 3-s − 5-s − 2·9-s − 5·11-s − 13-s − 15-s + 2·17-s + 19-s − 23-s + 25-s − 5·27-s + 29-s + 3·31-s − 5·33-s − 39-s − 2·41-s + 6·43-s + 2·45-s + 6·47-s − 7·49-s + 2·51-s − 6·53-s + 5·55-s + 57-s + 13·59-s + 8·61-s + 65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.50·11-s − 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s − 0.208·23-s + 1/5·25-s − 0.962·27-s + 0.185·29-s + 0.538·31-s − 0.870·33-s − 0.160·39-s − 0.312·41-s + 0.914·43-s + 0.298·45-s + 0.875·47-s − 49-s + 0.280·51-s − 0.824·53-s + 0.674·55-s + 0.132·57-s + 1.69·59-s + 1.02·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.469574990\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.469574990\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.014262238386210778708019646634, −7.65012317177265346704764622156, −6.80056013065565577813768631033, −5.78406302813653450456910971460, −5.26087365342507700534282251946, −4.42672519433363504977329230213, −3.44050281269323280716872192123, −2.82921648426784404958195849422, −2.11553893060718129978053901891, −0.58748963489756285552253104836,
0.58748963489756285552253104836, 2.11553893060718129978053901891, 2.82921648426784404958195849422, 3.44050281269323280716872192123, 4.42672519433363504977329230213, 5.26087365342507700534282251946, 5.78406302813653450456910971460, 6.80056013065565577813768631033, 7.65012317177265346704764622156, 8.014262238386210778708019646634