L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 4·11-s + 4·13-s + 4·19-s + 4·21-s + 2·23-s + 4·27-s + 2·29-s − 8·33-s + 4·37-s − 8·39-s + 2·41-s + 6·43-s + 6·47-s − 3·49-s − 4·53-s − 8·57-s + 12·59-s − 10·61-s − 2·63-s − 14·67-s − 4·69-s − 8·71-s + 8·73-s − 8·77-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 0.917·19-s + 0.872·21-s + 0.417·23-s + 0.769·27-s + 0.371·29-s − 1.39·33-s + 0.657·37-s − 1.28·39-s + 0.312·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.549·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s − 1.71·67-s − 0.481·69-s − 0.949·71-s + 0.936·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9261350608\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9261350608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−11.36915659252975788541915750236, −10.55414586727296810164329749828, −9.503819655627579392664602708544, −8.706139482166713221805533234674, −7.25328453174681220571113045442, −6.29510371522206160628039839956, −5.78805678883551867629350424855, −4.43125275610284135031254936982, −3.22911141285494921237059293026, −1.03411725818204990598413192839,
1.03411725818204990598413192839, 3.22911141285494921237059293026, 4.43125275610284135031254936982, 5.78805678883551867629350424855, 6.29510371522206160628039839956, 7.25328453174681220571113045442, 8.706139482166713221805533234674, 9.503819655627579392664602708544, 10.55414586727296810164329749828, 11.36915659252975788541915750236