| L(s) = 1 | − 2.21·3-s + 3.59·5-s + 7-s + 1.90·9-s + 3.52·11-s − 0.622·13-s − 7.95·15-s + 6.83·17-s − 2.21·21-s − 23-s + 7.90·25-s + 2.42·27-s + 1.09·29-s − 5.45·31-s − 7.80·33-s + 3.59·35-s + 7.37·37-s + 1.37·39-s + 9.61·41-s − 6.23·43-s + 6.83·45-s − 2.83·47-s + 49-s − 15.1·51-s − 13.6·53-s + 12.6·55-s + 13.2·59-s + ⋯ |
| L(s) = 1 | − 1.27·3-s + 1.60·5-s + 0.377·7-s + 0.634·9-s + 1.06·11-s − 0.172·13-s − 2.05·15-s + 1.65·17-s − 0.483·21-s − 0.208·23-s + 1.58·25-s + 0.467·27-s + 0.203·29-s − 0.980·31-s − 1.35·33-s + 0.607·35-s + 1.21·37-s + 0.220·39-s + 1.50·41-s − 0.950·43-s + 1.01·45-s − 0.413·47-s + 0.142·49-s − 2.11·51-s − 1.86·53-s + 1.70·55-s + 1.72·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.868379797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.868379797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 + 0.622T + 13T^{2} \) |
| 17 | \( 1 - 6.83T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 2.83T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 - 8.85T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 6.28T + 79T^{2} \) |
| 83 | \( 1 - 2.62T + 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−9.184233795949049350250281939154, −8.097287382502246806578869275161, −7.10147544859629610857348003562, −6.26488549489406664689472799243, −5.83011562414430700221985310297, −5.26196822787889671411518084890, −4.39506367102097691367442854388, −3.08582748284745616543515079856, −1.79240347289413001052110741592, −1.00583896859833268058079723988,
1.00583896859833268058079723988, 1.79240347289413001052110741592, 3.08582748284745616543515079856, 4.39506367102097691367442854388, 5.26196822787889671411518084890, 5.83011562414430700221985310297, 6.26488549489406664689472799243, 7.10147544859629610857348003562, 8.097287382502246806578869275161, 9.184233795949049350250281939154
