L(s) = 1 | − 2·4-s + 3·5-s + 7-s − 4·13-s + 4·16-s + 3·17-s + 2·19-s − 6·20-s + 4·25-s − 2·28-s + 6·29-s + 2·31-s + 3·35-s − 10·37-s − 6·41-s + 11·43-s + 3·47-s + 49-s + 8·52-s − 12·53-s + 3·59-s + 8·61-s − 8·64-s − 12·65-s + 5·67-s − 6·68-s + 12·71-s + ⋯ |
L(s) = 1 | − 4-s + 1.34·5-s + 0.377·7-s − 1.10·13-s + 16-s + 0.727·17-s + 0.458·19-s − 1.34·20-s + 4/5·25-s − 0.377·28-s + 1.11·29-s + 0.359·31-s + 0.507·35-s − 1.64·37-s − 0.937·41-s + 1.67·43-s + 0.437·47-s + 1/7·49-s + 1.10·52-s − 1.64·53-s + 0.390·59-s + 1.02·61-s − 64-s − 1.48·65-s + 0.610·67-s − 0.727·68-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.141285193\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.141285193\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.988558503480528851591159166502, −7.19741603700330141689904385521, −6.40499978060560693859412775955, −5.50903374703667124518338333851, −5.19719441753951767051998124967, −4.53697957520267012817369987236, −3.52343141720749364485357044732, −2.63643283341237998276585249813, −1.74086308734870954921413891975, −0.76208578730909250200409296360,
0.76208578730909250200409296360, 1.74086308734870954921413891975, 2.63643283341237998276585249813, 3.52343141720749364485357044732, 4.53697957520267012817369987236, 5.19719441753951767051998124967, 5.50903374703667124518338333851, 6.40499978060560693859412775955, 7.19741603700330141689904385521, 7.988558503480528851591159166502