| L(s) = 1 | − 2-s + 0.278·3-s + 4-s − 5-s − 0.278·6-s + 4.59·7-s − 8-s − 2.92·9-s + 10-s + 0.566·11-s + 0.278·12-s + 1.45·13-s − 4.59·14-s − 0.278·15-s + 16-s − 0.0142·17-s + 2.92·18-s + 7.85·19-s − 20-s + 1.27·21-s − 0.566·22-s + 2.81·23-s − 0.278·24-s + 25-s − 1.45·26-s − 1.64·27-s + 4.59·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.160·3-s + 0.5·4-s − 0.447·5-s − 0.113·6-s + 1.73·7-s − 0.353·8-s − 0.974·9-s + 0.316·10-s + 0.170·11-s + 0.0802·12-s + 0.404·13-s − 1.22·14-s − 0.0717·15-s + 0.250·16-s − 0.00345·17-s + 0.688·18-s + 1.80·19-s − 0.223·20-s + 0.278·21-s − 0.120·22-s + 0.586·23-s − 0.0567·24-s + 0.200·25-s − 0.286·26-s − 0.316·27-s + 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669611836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669611836\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.278T + 3T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 - 0.566T + 11T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 17 | \( 1 + 0.0142T + 17T^{2} \) |
| 19 | \( 1 - 7.85T + 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 - 4.06T + 31T^{2} \) |
| 37 | \( 1 + 0.140T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 + 1.47T + 43T^{2} \) |
| 47 | \( 1 + 8.80T + 47T^{2} \) |
| 53 | \( 1 + 0.608T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 - 8.63T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 7.41T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 8.77T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 0.629T + 97T^{2} \) |
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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
−8.360894129036923786663056935959, −7.86344807026024965069342567767, −7.35827270664395789996776804784, −6.32731583327990403862389621060, −5.35781161121132124520329446668, −4.87760065699869363588489975648, −3.68997990949679237076722745669, −2.86001999077204861960239510593, −1.75832736121523843384863739143, −0.863676544728277251893939428316,
0.863676544728277251893939428316, 1.75832736121523843384863739143, 2.86001999077204861960239510593, 3.68997990949679237076722745669, 4.87760065699869363588489975648, 5.35781161121132124520329446668, 6.32731583327990403862389621060, 7.35827270664395789996776804784, 7.86344807026024965069342567767, 8.360894129036923786663056935959
