| L(s) = 1 | − 0.678·3-s − 2.56·5-s − 4.63·7-s − 2.53·9-s + 0.0824i·11-s + 5.74i·13-s + 1.73·15-s − 2.55i·17-s − 4.26i·19-s + 3.14·21-s + (−2.75 + 3.92i)23-s + 1.56·25-s + 3.76·27-s + 4.77i·29-s − 6.46i·31-s + ⋯ |
| L(s) = 1 | − 0.391·3-s − 1.14·5-s − 1.75·7-s − 0.846·9-s + 0.0248i·11-s + 1.59i·13-s + 0.449·15-s − 0.619i·17-s − 0.978i·19-s + 0.686·21-s + (−0.574 + 0.818i)23-s + 0.312·25-s + 0.723·27-s + 0.886i·29-s − 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4313761626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4313761626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (2.75 - 3.92i)T \) |
| good | 3 | \( 1 + 0.678T + 3T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 - 0.0824iT - 11T^{2} \) |
| 13 | \( 1 - 5.74iT - 13T^{2} \) |
| 17 | \( 1 + 2.55iT - 17T^{2} \) |
| 19 | \( 1 + 4.26iT - 19T^{2} \) |
| 29 | \( 1 - 4.77iT - 29T^{2} \) |
| 31 | \( 1 + 6.46iT - 31T^{2} \) |
| 37 | \( 1 - 7.41T + 37T^{2} \) |
| 41 | \( 1 + 4.66T + 41T^{2} \) |
| 43 | \( 1 + 0.420iT - 43T^{2} \) |
| 47 | \( 1 + 9.59iT - 47T^{2} \) |
| 53 | \( 1 - 9.22T + 53T^{2} \) |
| 59 | \( 1 - 2.10T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 0.188iT - 67T^{2} \) |
| 71 | \( 1 + 3.17iT - 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 7.33iT - 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 3.42iT - 97T^{2} \) |
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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
−9.294852876649021276950684699686, −8.909367599717089757433103180758, −7.71750902893991403446058367169, −6.89847522447175377270908104190, −6.38757204124899128288512200346, −5.36535992181202184036191021244, −4.20712271459880410139571019603, −3.49980958348976524483809493963, −2.51292361006592140088585988413, −0.37653051750006200723747172599,
0.53249584437322287735522303451, 2.82633523986182539145404048204, 3.41219065481470964481470302051, 4.32022592344397596768843508660, 5.77272174417881270003436687015, 6.06196669634590385757317867534, 7.11094097750896366434921868443, 8.090227846141441915659481023723, 8.498713125406863063290711390392, 9.735855859224344068245012632837
