L(s) = 1 | + 2·3-s + 4·5-s + 9-s − 3·11-s + 5·13-s + 8·15-s − 3·17-s + 2·19-s − 23-s + 11·25-s − 4·27-s + 6·29-s − 31-s − 6·33-s + 10·39-s + 5·41-s + 43-s + 4·45-s + 4·47-s − 7·49-s − 6·51-s + 5·53-s − 12·55-s + 4·57-s + 12·59-s − 2·61-s + 20·65-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.78·5-s + 1/3·9-s − 0.904·11-s + 1.38·13-s + 2.06·15-s − 0.727·17-s + 0.458·19-s − 0.208·23-s + 11/5·25-s − 0.769·27-s + 1.11·29-s − 0.179·31-s − 1.04·33-s + 1.60·39-s + 0.780·41-s + 0.152·43-s + 0.596·45-s + 0.583·47-s − 49-s − 0.840·51-s + 0.686·53-s − 1.61·55-s + 0.529·57-s + 1.56·59-s − 0.256·61-s + 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.866947450\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.866947450\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.717465909637325225693009204963, −8.424036461530773064911189485868, −7.37908773560096266400471473449, −6.40766197175942745490156512688, −5.81506625354326521208901179447, −5.04783228725436586348732804822, −3.86026229009991354251089428426, −2.80098979618964440556595833817, −2.31845882511644578775258854916, −1.29567362974576238576228685946,
1.29567362974576238576228685946, 2.31845882511644578775258854916, 2.80098979618964440556595833817, 3.86026229009991354251089428426, 5.04783228725436586348732804822, 5.81506625354326521208901179447, 6.40766197175942745490156512688, 7.37908773560096266400471473449, 8.424036461530773064911189485868, 8.717465909637325225693009204963