| L(s) = 1 | − 17.5·3-s + 49·7-s + 64.7·9-s − 444.·11-s − 705.·13-s + 363.·17-s − 280.·19-s − 859.·21-s − 2.95e3·23-s + 3.12e3·27-s + 5.09e3·29-s − 4.39e3·31-s + 7.79e3·33-s − 1.11e4·37-s + 1.23e4·39-s − 2.00e4·41-s + 4.09e3·43-s + 2.01e4·47-s + 2.40e3·49-s − 6.37e3·51-s − 1.86e4·53-s + 4.91e3·57-s − 3.68e4·59-s − 2.38e4·61-s + 3.17e3·63-s + 2.63e4·67-s + 5.19e4·69-s + ⋯ |
| L(s) = 1 | − 1.12·3-s + 0.377·7-s + 0.266·9-s − 1.10·11-s − 1.15·13-s + 0.305·17-s − 0.178·19-s − 0.425·21-s − 1.16·23-s + 0.825·27-s + 1.12·29-s − 0.821·31-s + 1.24·33-s − 1.34·37-s + 1.30·39-s − 1.86·41-s + 0.337·43-s + 1.32·47-s + 0.142·49-s − 0.343·51-s − 0.913·53-s + 0.200·57-s − 1.37·59-s − 0.819·61-s + 0.100·63-s + 0.716·67-s + 1.31·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4508627132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4508627132\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 17.5T + 243T^{2} \) |
| 11 | \( 1 + 444.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 705.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 363.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 280.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.95e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.09e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.86e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.65e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.59e3T + 8.58e9T^{2} \) |
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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
−10.10716855865305126932243446581, −8.761014044801854912090588940490, −7.84071147251677510855585663272, −7.02061980613042462819124923757, −5.94396083563639225776434459477, −5.21838381355349977618491643502, −4.54792998191904483231233347270, −3.01684987750985468153075558252, −1.83038117537204298642594176778, −0.32135166992628112006299420690,
0.32135166992628112006299420690, 1.83038117537204298642594176778, 3.01684987750985468153075558252, 4.54792998191904483231233347270, 5.21838381355349977618491643502, 5.94396083563639225776434459477, 7.02061980613042462819124923757, 7.84071147251677510855585663272, 8.761014044801854912090588940490, 10.10716855865305126932243446581
