L(s) = 1 | + 2.69·2-s − 3-s + 5.26·4-s + 2.08·5-s − 2.69·6-s + 8.78·8-s + 9-s + 5.60·10-s + 3.71·11-s − 5.26·12-s + 6.98·13-s − 2.08·15-s + 13.1·16-s − 0.439·17-s + 2.69·18-s − 0.527·19-s + 10.9·20-s + 10.0·22-s − 7.76·23-s − 8.78·24-s − 0.670·25-s + 18.8·26-s − 27-s − 2.31·29-s − 5.60·30-s − 1.64·31-s + 17.8·32-s + ⋯ |
L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.63·4-s + 0.930·5-s − 1.10·6-s + 3.10·8-s + 0.333·9-s + 1.77·10-s + 1.11·11-s − 1.51·12-s + 1.93·13-s − 0.537·15-s + 3.28·16-s − 0.106·17-s + 0.635·18-s − 0.121·19-s + 2.44·20-s + 2.13·22-s − 1.61·23-s − 1.79·24-s − 0.134·25-s + 3.69·26-s − 0.192·27-s − 0.429·29-s − 1.02·30-s − 0.295·31-s + 3.16·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.393268312\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.393268312\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 - 6.98T + 13T^{2} \) |
| 17 | \( 1 + 0.439T + 17T^{2} \) |
| 19 | \( 1 + 0.527T + 19T^{2} \) |
| 23 | \( 1 + 7.76T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 - 9.81T + 37T^{2} \) |
| 43 | \( 1 + 8.88T + 43T^{2} \) |
| 47 | \( 1 + 9.76T + 47T^{2} \) |
| 53 | \( 1 - 1.90T + 53T^{2} \) |
| 59 | \( 1 - 9.36T + 59T^{2} \) |
| 61 | \( 1 + 1.50T + 61T^{2} \) |
| 67 | \( 1 - 1.04T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 - 3.01T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 9.86T + 89T^{2} \) |
| 97 | \( 1 - 5.45T + 97T^{2} \) |
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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
−7.81012444380464064430983395418, −6.68810738063918174167634032462, −6.40694613294569658337326105350, −5.83019638779546924876400816019, −5.43255989588696493991907185905, −4.22846369042187182863258287177, −3.99857864058834633682175013932, −3.08890126228408164816503625611, −1.88270931436110004607930638424, −1.42209026368258556837820450263,
1.42209026368258556837820450263, 1.88270931436110004607930638424, 3.08890126228408164816503625611, 3.99857864058834633682175013932, 4.22846369042187182863258287177, 5.43255989588696493991907185905, 5.83019638779546924876400816019, 6.40694613294569658337326105350, 6.68810738063918174167634032462, 7.81012444380464064430983395418