L(s) = 1 | − 3·3-s + 16.2·7-s + 9·9-s + 40.2·11-s + 19.7·13-s + 83.0·17-s + 48.8·19-s − 48.6·21-s − 1.61·23-s − 27·27-s − 24.5·29-s + 12.4·31-s − 120.·33-s + 325.·37-s − 59.3·39-s − 242.·41-s − 367.·43-s + 204.·47-s − 80.2·49-s − 249.·51-s − 61.5·53-s − 146.·57-s + 112.·59-s + 477.·61-s + 145.·63-s − 558.·67-s + 4.83·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.875·7-s + 0.333·9-s + 1.10·11-s + 0.422·13-s + 1.18·17-s + 0.589·19-s − 0.505·21-s − 0.0146·23-s − 0.192·27-s − 0.157·29-s + 0.0719·31-s − 0.636·33-s + 1.44·37-s − 0.243·39-s − 0.923·41-s − 1.30·43-s + 0.634·47-s − 0.233·49-s − 0.683·51-s − 0.159·53-s − 0.340·57-s + 0.247·59-s + 1.00·61-s + 0.291·63-s − 1.01·67-s + 0.00844·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.369189929\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.369189929\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 16.2T + 343T^{2} \) |
| 11 | \( 1 - 40.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.61T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 61.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 477.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 558.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 558.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 96.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−9.478131657047385873086771160621, −8.489527145532970048176229211555, −7.73911056024936449867241792804, −6.84372110389886155498715598437, −5.95799084770097863095443770661, −5.17382705644969217853794916366, −4.25407457691096052783135073048, −3.28569540606277144488958352111, −1.69619500963758275343443964988, −0.892085465030633368401629993636,
0.892085465030633368401629993636, 1.69619500963758275343443964988, 3.28569540606277144488958352111, 4.25407457691096052783135073048, 5.17382705644969217853794916366, 5.95799084770097863095443770661, 6.84372110389886155498715598437, 7.73911056024936449867241792804, 8.489527145532970048176229211555, 9.478131657047385873086771160621