| L(s) = 1 | − 3.04·3-s − 1.92·7-s + 6.24·9-s − 5.98·11-s + 1.61·13-s − 17-s + 7.70·19-s + 5.84·21-s − 2.09·23-s − 9.86·27-s − 3.10·29-s + 7.68·31-s + 18.1·33-s − 11.7·37-s − 4.92·39-s − 1.62·41-s − 2.98·43-s + 6.60·47-s − 3.30·49-s + 3.04·51-s − 7.94·53-s − 23.4·57-s − 9.10·59-s + 10.0·61-s − 12.0·63-s − 8.48·67-s + 6.38·69-s + ⋯ |
| L(s) = 1 | − 1.75·3-s − 0.726·7-s + 2.08·9-s − 1.80·11-s + 0.448·13-s − 0.242·17-s + 1.76·19-s + 1.27·21-s − 0.437·23-s − 1.89·27-s − 0.575·29-s + 1.38·31-s + 3.16·33-s − 1.92·37-s − 0.788·39-s − 0.253·41-s − 0.454·43-s + 0.963·47-s − 0.472·49-s + 0.425·51-s − 1.09·53-s − 3.10·57-s − 1.18·59-s + 1.29·61-s − 1.51·63-s − 1.03·67-s + 0.768·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4253878923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4253878923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 3.04T + 3T^{2} \) |
| 7 | \( 1 + 1.92T + 7T^{2} \) |
| 11 | \( 1 + 5.98T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 1.62T + 41T^{2} \) |
| 43 | \( 1 + 2.98T + 43T^{2} \) |
| 47 | \( 1 - 6.60T + 47T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 8.64T + 71T^{2} \) |
| 73 | \( 1 - 0.257T + 73T^{2} \) |
| 79 | \( 1 + 5.13T + 79T^{2} \) |
| 83 | \( 1 - 1.84T + 83T^{2} \) |
| 89 | \( 1 - 9.10T + 89T^{2} \) |
| 97 | \( 1 - 6.64T + 97T^{2} \) |
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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
−7.66492868355221930777133856142, −7.21765213516434097634607201843, −6.36739304780903642076105878953, −5.85339999387176625550619576219, −5.17064537647965148443687830908, −4.79629536542733410902191266901, −3.62508180557397755212006040704, −2.81706144470412212712320886555, −1.50790835930636604804072202727, −0.37494531260441389820383693024,
0.37494531260441389820383693024, 1.50790835930636604804072202727, 2.81706144470412212712320886555, 3.62508180557397755212006040704, 4.79629536542733410902191266901, 5.17064537647965148443687830908, 5.85339999387176625550619576219, 6.36739304780903642076105878953, 7.21765213516434097634607201843, 7.66492868355221930777133856142
