| L(s) = 1 | + 2-s − 0.919·3-s + 4-s − 5-s − 0.919·6-s + 7-s + 8-s − 2.15·9-s − 10-s − 0.919·12-s − 3.72·13-s + 14-s + 0.919·15-s + 16-s + 1.31·17-s − 2.15·18-s + 2.70·19-s − 20-s − 0.919·21-s − 2·23-s − 0.919·24-s + 25-s − 3.72·26-s + 4.73·27-s + 28-s + 2.95·29-s + 0.919·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.530·3-s + 0.5·4-s − 0.447·5-s − 0.375·6-s + 0.377·7-s + 0.353·8-s − 0.718·9-s − 0.316·10-s − 0.265·12-s − 1.03·13-s + 0.267·14-s + 0.237·15-s + 0.250·16-s + 0.319·17-s − 0.507·18-s + 0.621·19-s − 0.223·20-s − 0.200·21-s − 0.417·23-s − 0.187·24-s + 0.200·25-s − 0.730·26-s + 0.911·27-s + 0.188·28-s + 0.547·29-s + 0.167·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.976830205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976830205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.919T + 3T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 2.95T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 + 8.04T + 37T^{2} \) |
| 41 | \( 1 + 3.24T + 41T^{2} \) |
| 43 | \( 1 + 6.48T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 2.83T + 59T^{2} \) |
| 61 | \( 1 + 1.44T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 - 8.03T + 71T^{2} \) |
| 73 | \( 1 - 1.33T + 73T^{2} \) |
| 79 | \( 1 - 7.72T + 79T^{2} \) |
| 83 | \( 1 + 0.285T + 83T^{2} \) |
| 89 | \( 1 + 8.14T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.74394520533425307838361722079, −6.92142599658560554389912518449, −6.40451018736395434327431133727, −5.49909124357501784793813115330, −5.03609861527455050110544065850, −4.48659605462219639862144507841, −3.44804057570829548576317031608, −2.85908030240174506612132084873, −1.87630047434555690430786727106, −0.61977358199306326016596966969,
0.61977358199306326016596966969, 1.87630047434555690430786727106, 2.85908030240174506612132084873, 3.44804057570829548576317031608, 4.48659605462219639862144507841, 5.03609861527455050110544065850, 5.49909124357501784793813115330, 6.40451018736395434327431133727, 6.92142599658560554389912518449, 7.74394520533425307838361722079
