| L(s) = 1 | + 0.809·3-s + 3.65·7-s − 2.34·9-s + 2.15·11-s + 6.40·13-s + 3.77·17-s − 6.39·19-s + 2.95·21-s − 1.51·23-s − 4.32·27-s + 29-s + 7.48·31-s + 1.74·33-s − 2.15·37-s + 5.17·39-s + 7.22·41-s + 8.95·43-s − 3.84·47-s + 6.32·49-s + 3.05·51-s − 9.89·53-s − 5.16·57-s + 1.36·59-s − 7.82·61-s − 8.56·63-s + 2.29·67-s − 1.22·69-s + ⋯ |
| L(s) = 1 | + 0.467·3-s + 1.37·7-s − 0.781·9-s + 0.650·11-s + 1.77·13-s + 0.915·17-s − 1.46·19-s + 0.644·21-s − 0.316·23-s − 0.832·27-s + 0.185·29-s + 1.34·31-s + 0.303·33-s − 0.354·37-s + 0.829·39-s + 1.12·41-s + 1.36·43-s − 0.561·47-s + 0.903·49-s + 0.427·51-s − 1.35·53-s − 0.684·57-s + 0.177·59-s − 1.00·61-s − 1.07·63-s + 0.279·67-s − 0.147·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.776142966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.776142966\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.809T + 3T^{2} \) |
| 7 | \( 1 - 3.65T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 + 6.39T + 19T^{2} \) |
| 23 | \( 1 + 1.51T + 23T^{2} \) |
| 31 | \( 1 - 7.48T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 - 8.95T + 43T^{2} \) |
| 47 | \( 1 + 3.84T + 47T^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 - 1.36T + 59T^{2} \) |
| 61 | \( 1 + 7.82T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 + 3.13T + 71T^{2} \) |
| 73 | \( 1 + 5.90T + 73T^{2} \) |
| 79 | \( 1 + 6.34T + 79T^{2} \) |
| 83 | \( 1 + 8.30T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.51T + 97T^{2} \) |
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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
−8.560423018843002380190452031478, −8.241860933155986231346318701466, −7.52776424765547387680294817280, −6.19528555959584551585663328917, −5.95919280719205725113225410319, −4.72667166143513485983728850689, −4.03572972324856849044613084382, −3.13420310589914694288988783100, −2.00109217445985668205961918381, −1.09756841811716100646712719921,
1.09756841811716100646712719921, 2.00109217445985668205961918381, 3.13420310589914694288988783100, 4.03572972324856849044613084382, 4.72667166143513485983728850689, 5.95919280719205725113225410319, 6.19528555959584551585663328917, 7.52776424765547387680294817280, 8.241860933155986231346318701466, 8.560423018843002380190452031478
