| L(s) = 1 | − 3-s + 3.63·5-s + 2.74·7-s + 9-s + 3.30·11-s + 5.31·13-s − 3.63·15-s + 3.11·17-s − 4.91·19-s − 2.74·21-s − 1.16·23-s + 8.22·25-s − 27-s − 3.53·29-s + 2.72·31-s − 3.30·33-s + 9.97·35-s + 2.05·37-s − 5.31·39-s + 4.50·41-s + 3.85·43-s + 3.63·45-s + 3.91·47-s + 0.521·49-s − 3.11·51-s + 9.24·53-s + 12.0·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.62·5-s + 1.03·7-s + 0.333·9-s + 0.997·11-s + 1.47·13-s − 0.938·15-s + 0.755·17-s − 1.12·19-s − 0.598·21-s − 0.243·23-s + 1.64·25-s − 0.192·27-s − 0.655·29-s + 0.489·31-s − 0.575·33-s + 1.68·35-s + 0.337·37-s − 0.851·39-s + 0.704·41-s + 0.588·43-s + 0.542·45-s + 0.570·47-s + 0.0745·49-s − 0.436·51-s + 1.26·53-s + 1.62·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.266294453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.266294453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 - 3.63T + 5T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 2.72T + 31T^{2} \) |
| 37 | \( 1 - 2.05T + 37T^{2} \) |
| 41 | \( 1 - 4.50T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 - 3.91T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 + 8.59T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 7.25T + 97T^{2} \) |
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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
−8.228791254951184593997189165396, −7.21100920295651392063387223693, −6.39280339475091555638258709534, −5.90847216227626380341864092375, −5.49644493399331927728945871984, −4.47813037505713050773609155507, −3.85643731073074770707214762883, −2.53169770866294442716576807857, −1.58200383339784564815667169182, −1.16031994171382652659125582777,
1.16031994171382652659125582777, 1.58200383339784564815667169182, 2.53169770866294442716576807857, 3.85643731073074770707214762883, 4.47813037505713050773609155507, 5.49644493399331927728945871984, 5.90847216227626380341864092375, 6.39280339475091555638258709534, 7.21100920295651392063387223693, 8.228791254951184593997189165396
