| L(s) = 1 | + 2-s + 4-s + 3.14·5-s − 7-s + 8-s + 3.14·10-s − 0.726·11-s + 2.36·13-s − 14-s + 16-s + 2.77·17-s − 1.50·19-s + 3.14·20-s − 0.726·22-s − 23-s + 4.86·25-s + 2.36·26-s − 28-s + 7.86·29-s − 0.778·31-s + 32-s + 2.77·34-s − 3.14·35-s + 5.86·37-s − 1.50·38-s + 3.14·40-s + 1.58·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.40·5-s − 0.377·7-s + 0.353·8-s + 0.993·10-s − 0.219·11-s + 0.655·13-s − 0.267·14-s + 0.250·16-s + 0.673·17-s − 0.345·19-s + 0.702·20-s − 0.154·22-s − 0.208·23-s + 0.973·25-s + 0.463·26-s − 0.188·28-s + 1.46·29-s − 0.139·31-s + 0.176·32-s + 0.476·34-s − 0.530·35-s + 0.964·37-s − 0.244·38-s + 0.496·40-s + 0.247·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.885021715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.885021715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.14T + 5T^{2} \) |
| 11 | \( 1 + 0.726T + 11T^{2} \) |
| 13 | \( 1 - 2.36T + 13T^{2} \) |
| 17 | \( 1 - 2.77T + 17T^{2} \) |
| 19 | \( 1 + 1.50T + 19T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 0.778T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + 2.64T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.55T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 - 9.29T + 71T^{2} \) |
| 73 | \( 1 + 1.00T + 73T^{2} \) |
| 79 | \( 1 - 2.72T + 79T^{2} \) |
| 83 | \( 1 + 7.78T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−8.754321468551801002162142078799, −8.043018369137530039350956875165, −6.91409990776087408595891883458, −6.32301376061698191478195807631, −5.72361960408451910680754910021, −5.05244758763763715692628298545, −4.03665422067426175057442861656, −3.03799381185651346704844186676, −2.25647968242753906771195353186, −1.18948940952964245253188440334,
1.18948940952964245253188440334, 2.25647968242753906771195353186, 3.03799381185651346704844186676, 4.03665422067426175057442861656, 5.05244758763763715692628298545, 5.72361960408451910680754910021, 6.32301376061698191478195807631, 6.91409990776087408595891883458, 8.043018369137530039350956875165, 8.754321468551801002162142078799
