| L(s) = 1 | − 2-s − 0.475·3-s + 4-s + 1.98·5-s + 0.475·6-s + 3.40·7-s − 8-s − 2.77·9-s − 1.98·10-s − 2.07·11-s − 0.475·12-s − 5.67·13-s − 3.40·14-s − 0.944·15-s + 16-s + 2.55·17-s + 2.77·18-s + 3.34·19-s + 1.98·20-s − 1.61·21-s + 2.07·22-s + 7.08·23-s + 0.475·24-s − 1.05·25-s + 5.67·26-s + 2.74·27-s + 3.40·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.274·3-s + 0.5·4-s + 0.888·5-s + 0.194·6-s + 1.28·7-s − 0.353·8-s − 0.924·9-s − 0.628·10-s − 0.626·11-s − 0.137·12-s − 1.57·13-s − 0.910·14-s − 0.243·15-s + 0.250·16-s + 0.620·17-s + 0.653·18-s + 0.768·19-s + 0.444·20-s − 0.353·21-s + 0.443·22-s + 1.47·23-s + 0.0970·24-s − 0.210·25-s + 1.11·26-s + 0.528·27-s + 0.643·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.415865154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.415865154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 0.475T + 3T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 7 | \( 1 - 3.40T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 - 7.08T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 37 | \( 1 + 9.15T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 7.02T + 43T^{2} \) |
| 47 | \( 1 - 6.77T + 47T^{2} \) |
| 53 | \( 1 - 2.36T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 - 2.02T + 61T^{2} \) |
| 67 | \( 1 - 0.372T + 67T^{2} \) |
| 71 | \( 1 + 6.22T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
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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
−7.891617798986728934481041637720, −7.60538487341987154780335485326, −6.84826960900790130848512747602, −5.73710031703871061260853289249, −5.27788234718504024192123449327, −4.90665128905675647477296635752, −3.38196007719362966945023415809, −2.44910115044668321166421645578, −1.87849028423846671408630961865, −0.69579513183364456199143143517,
0.69579513183364456199143143517, 1.87849028423846671408630961865, 2.44910115044668321166421645578, 3.38196007719362966945023415809, 4.90665128905675647477296635752, 5.27788234718504024192123449327, 5.73710031703871061260853289249, 6.84826960900790130848512747602, 7.60538487341987154780335485326, 7.891617798986728934481041637720
