| L(s) = 1 | + 2-s − 3-s + 4-s − 3.42·5-s − 6-s − 1.25·7-s + 8-s + 9-s − 3.42·10-s − 2.48·11-s − 12-s − 5.96·13-s − 1.25·14-s + 3.42·15-s + 16-s − 5.90·17-s + 18-s + 0.0494·19-s − 3.42·20-s + 1.25·21-s − 2.48·22-s − 23-s − 24-s + 6.71·25-s − 5.96·26-s − 27-s − 1.25·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.53·5-s − 0.408·6-s − 0.473·7-s + 0.353·8-s + 0.333·9-s − 1.08·10-s − 0.747·11-s − 0.288·12-s − 1.65·13-s − 0.334·14-s + 0.883·15-s + 0.250·16-s − 1.43·17-s + 0.235·18-s + 0.0113·19-s − 0.765·20-s + 0.273·21-s − 0.528·22-s − 0.208·23-s − 0.204·24-s + 1.34·25-s − 1.16·26-s − 0.192·27-s − 0.236·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7108217192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7108217192\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 + 5.96T + 13T^{2} \) |
| 17 | \( 1 + 5.90T + 17T^{2} \) |
| 19 | \( 1 - 0.0494T + 19T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 6.37T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 + 0.286T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 - 4.48T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 0.796T + 89T^{2} \) |
| 97 | \( 1 + 0.811T + 97T^{2} \) |
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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
−8.112310332834218977072934048079, −7.57672785419008241617813953570, −6.98502657390406066823172513517, −6.27827577487881850756498123689, −5.27013744827923693306452894101, −4.55287977913340675055817837799, −4.14310426609831373419575499539, −3.04229005679287920766225697693, −2.29302335023659879196793710071, −0.41601733269579951668410811674,
0.41601733269579951668410811674, 2.29302335023659879196793710071, 3.04229005679287920766225697693, 4.14310426609831373419575499539, 4.55287977913340675055817837799, 5.27013744827923693306452894101, 6.27827577487881850756498123689, 6.98502657390406066823172513517, 7.57672785419008241617813953570, 8.112310332834218977072934048079
