| L(s) = 1 | + i·2-s − 4-s + (−0.0951 − 2.64i)7-s − i·8-s − 5.28i·11-s + 2.19i·13-s + (2.64 − 0.0951i)14-s + 16-s − 1.04·17-s − 6.43i·19-s + 5.28·22-s + 7.47i·23-s − 2.19·26-s + (0.0951 + 2.64i)28-s − 7.47i·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.0359 − 0.999i)7-s − 0.353i·8-s − 1.59i·11-s + 0.607i·13-s + (0.706 − 0.0254i)14-s + 0.250·16-s − 0.253·17-s − 1.47i·19-s + 1.12·22-s + 1.55i·23-s − 0.429·26-s + (0.0179 + 0.499i)28-s − 1.38i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6365975010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6365975010\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.0951 + 2.64i)T \) |
| good | 11 | \( 1 + 5.28iT - 11T^{2} \) |
| 13 | \( 1 - 2.19iT - 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 + 6.43iT - 19T^{2} \) |
| 23 | \( 1 - 7.47iT - 23T^{2} \) |
| 29 | \( 1 + 7.47iT - 29T^{2} \) |
| 31 | \( 1 - 9.09iT - 31T^{2} \) |
| 37 | \( 1 - 0.855T + 37T^{2} \) |
| 41 | \( 1 + 2.19T + 41T^{2} \) |
| 43 | \( 1 - 0.954T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 3.09iT - 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.05iT - 61T^{2} \) |
| 67 | \( 1 + 5.33T + 67T^{2} \) |
| 71 | \( 1 - 6.43iT - 71T^{2} \) |
| 73 | \( 1 - 4.57iT - 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 4.38T + 83T^{2} \) |
| 89 | \( 1 + 4.28T + 89T^{2} \) |
| 97 | \( 1 + 11.8iT - 97T^{2} \) |
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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
−8.290255093040938746659901371111, −7.69027024983537908468194541993, −6.79925429399193409078747454876, −6.39392440730882804582665070598, −5.38818361919009060137774198582, −4.65740073921136099336954018022, −3.73642836469040030198739099510, −3.01653851854856646531370819900, −1.37632763665292059651761729974, −0.19873983003084499707183127570,
1.58940311492578859552958713557, 2.33137771344974794232911307186, 3.21153580476244868162254556021, 4.30630665653904665578194173778, 4.93521868377378313801878558992, 5.83021664595739277194153467766, 6.58762878306877140251035692084, 7.66862621559015938256391992977, 8.245493333362655473362396332674, 9.097417305828548760651132973326
