| L(s) = 1 | − 3.05i·5-s + 1.35i·7-s + 5.12·11-s + 5.32·13-s − 4.54i·17-s − 1.11i·19-s + 0.226·23-s − 4.34·25-s − 3.49i·29-s + 0.0119i·31-s + 4.14·35-s − 37-s − 5.56i·41-s + 6.47i·43-s + 6.43·47-s + ⋯ |
| L(s) = 1 | − 1.36i·5-s + 0.512i·7-s + 1.54·11-s + 1.47·13-s − 1.10i·17-s − 0.254i·19-s + 0.0471·23-s − 0.869·25-s − 0.648i·29-s + 0.00213i·31-s + 0.700·35-s − 0.164·37-s − 0.869i·41-s + 0.987i·43-s + 0.938·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.464903433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.464903433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 3.05iT - 5T^{2} \) |
| 7 | \( 1 - 1.35iT - 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 5.32T + 13T^{2} \) |
| 17 | \( 1 + 4.54iT - 17T^{2} \) |
| 19 | \( 1 + 1.11iT - 19T^{2} \) |
| 23 | \( 1 - 0.226T + 23T^{2} \) |
| 29 | \( 1 + 3.49iT - 29T^{2} \) |
| 31 | \( 1 - 0.0119iT - 31T^{2} \) |
| 41 | \( 1 + 5.56iT - 41T^{2} \) |
| 43 | \( 1 - 6.47iT - 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 - 4.20iT - 53T^{2} \) |
| 59 | \( 1 - 4.13T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 - 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 - 1.62T + 73T^{2} \) |
| 79 | \( 1 - 3.25iT - 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−8.370693760038716134216969496880, −7.34225845751754374106373805825, −6.53387017225026601338776938835, −5.82566916974757873671609848119, −5.20470317389339104868666445455, −4.26663906807021484594037824616, −3.82163933689348605288402651500, −2.61001193664111947964499647759, −1.41523110385281105020173933584, −0.810909816587391651101247901508,
1.11701440413121301667238861416, 1.97981485390650300804004622352, 3.31048241267796413506178877030, 3.67539190076274010169392109129, 4.35066210256652152558976798517, 5.75334761035536033950527441470, 6.28387541198898194094178032278, 6.82130738690033609553259346277, 7.39659139914997381530315361542, 8.416814289593978904754071824095
