| L(s) = 1 | + 1.27·3-s + 4.13·5-s − 1.36·9-s + 1.53·11-s + 0.0160·13-s + 5.29·15-s − 17-s + 0.519·19-s + 0.958·23-s + 12.1·25-s − 5.58·27-s + 5.79·29-s + 1.29·31-s + 1.96·33-s + 2.76·37-s + 0.0205·39-s − 0.769·41-s − 4.93·43-s − 5.64·45-s − 1.60·47-s − 1.27·51-s + 5.00·53-s + 6.36·55-s + 0.665·57-s + 5.50·59-s + 14.5·61-s + 0.0664·65-s + ⋯ |
| L(s) = 1 | + 0.738·3-s + 1.85·5-s − 0.454·9-s + 0.463·11-s + 0.00445·13-s + 1.36·15-s − 0.242·17-s + 0.119·19-s + 0.199·23-s + 2.42·25-s − 1.07·27-s + 1.07·29-s + 0.232·31-s + 0.342·33-s + 0.455·37-s + 0.00329·39-s − 0.120·41-s − 0.752·43-s − 0.840·45-s − 0.234·47-s − 0.179·51-s + 0.687·53-s + 0.857·55-s + 0.0880·57-s + 0.716·59-s + 1.86·61-s + 0.00824·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.539555007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.539555007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.27T + 3T^{2} \) |
| 5 | \( 1 - 4.13T + 5T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 - 0.0160T + 13T^{2} \) |
| 19 | \( 1 - 0.519T + 19T^{2} \) |
| 23 | \( 1 - 0.958T + 23T^{2} \) |
| 29 | \( 1 - 5.79T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 + 0.769T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 - 5.00T + 53T^{2} \) |
| 59 | \( 1 - 5.50T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 5.13T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.659477724848826089415854707325, −8.193631503982321038110989994388, −6.91897584985450433192522103168, −6.42912693009325012409174231199, −5.59991865772049512847707621172, −4.99476484466996759052207179900, −3.78898819833989986959674772892, −2.73951632506840970770739499870, −2.22998758619064200829577280762, −1.17040288994534890482874969269,
1.17040288994534890482874969269, 2.22998758619064200829577280762, 2.73951632506840970770739499870, 3.78898819833989986959674772892, 4.99476484466996759052207179900, 5.59991865772049512847707621172, 6.42912693009325012409174231199, 6.91897584985450433192522103168, 8.193631503982321038110989994388, 8.659477724848826089415854707325
