| L(s) = 1 | + 2-s − 3-s + 4-s − 3.78·5-s − 6-s + 3.57·7-s + 8-s + 9-s − 3.78·10-s + 2.34·11-s − 12-s + 4.66·13-s + 3.57·14-s + 3.78·15-s + 16-s − 1.44·17-s + 18-s − 1.40·19-s − 3.78·20-s − 3.57·21-s + 2.34·22-s − 23-s − 24-s + 9.34·25-s + 4.66·26-s − 27-s + 3.57·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.69·5-s − 0.408·6-s + 1.35·7-s + 0.353·8-s + 0.333·9-s − 1.19·10-s + 0.706·11-s − 0.288·12-s + 1.29·13-s + 0.954·14-s + 0.977·15-s + 0.250·16-s − 0.350·17-s + 0.235·18-s − 0.321·19-s − 0.846·20-s − 0.779·21-s + 0.499·22-s − 0.208·23-s − 0.204·24-s + 1.86·25-s + 0.915·26-s − 0.192·27-s + 0.675·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.307489279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.307489279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 3.78T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 + 1.40T + 19T^{2} \) |
| 31 | \( 1 - 1.30T + 31T^{2} \) |
| 37 | \( 1 - 8.01T + 37T^{2} \) |
| 41 | \( 1 + 7.66T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 - 0.554T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 + 0.342T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 0.445T + 73T^{2} \) |
| 79 | \( 1 + 5.56T + 79T^{2} \) |
| 83 | \( 1 - 8.84T + 83T^{2} \) |
| 89 | \( 1 - 7.99T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−8.202360613154487162192422962891, −7.80625067621528711616615952810, −6.83201704647473664484900793117, −6.32208835229419961039127583041, −5.22465335773459744901128997490, −4.59013464669652377240529275375, −3.98582485600240340930660886389, −3.40340792365937703475036562351, −1.87910477667840933836565232196, −0.847626420433911442941772139968,
0.847626420433911442941772139968, 1.87910477667840933836565232196, 3.40340792365937703475036562351, 3.98582485600240340930660886389, 4.59013464669652377240529275375, 5.22465335773459744901128997490, 6.32208835229419961039127583041, 6.83201704647473664484900793117, 7.80625067621528711616615952810, 8.202360613154487162192422962891
