| L(s) = 1 | + 1.74·2-s − 3-s + 1.05·4-s − 2.47·5-s − 1.74·6-s + 4.46·7-s − 1.65·8-s + 9-s − 4.33·10-s + 3.56·11-s − 1.05·12-s − 3.22·13-s + 7.79·14-s + 2.47·15-s − 4.99·16-s + 17-s + 1.74·18-s + 7.76·19-s − 2.60·20-s − 4.46·21-s + 6.22·22-s + 0.531·23-s + 1.65·24-s + 1.14·25-s − 5.63·26-s − 27-s + 4.68·28-s + ⋯ |
| L(s) = 1 | + 1.23·2-s − 0.577·3-s + 0.525·4-s − 1.10·5-s − 0.713·6-s + 1.68·7-s − 0.586·8-s + 0.333·9-s − 1.36·10-s + 1.07·11-s − 0.303·12-s − 0.894·13-s + 2.08·14-s + 0.640·15-s − 1.24·16-s + 0.242·17-s + 0.411·18-s + 1.78·19-s − 0.582·20-s − 0.973·21-s + 1.32·22-s + 0.110·23-s + 0.338·24-s + 0.229·25-s − 1.10·26-s − 0.192·27-s + 0.886·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.785639503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.785639503\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 1.74T + 2T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 19 | \( 1 - 7.76T + 19T^{2} \) |
| 23 | \( 1 - 0.531T + 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 + 7.18T + 41T^{2} \) |
| 43 | \( 1 - 2.94T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 - 8.50T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 1.97T + 61T^{2} \) |
| 67 | \( 1 + 6.31T + 67T^{2} \) |
| 71 | \( 1 + 2.42T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 83 | \( 1 - 3.49T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.241359568143392443287980136853, −7.44503712049484423018361292011, −7.08596646204159274256067141042, −5.87742396321663981472484837893, −5.21116779224336242323026244517, −4.74640618715184076250855443997, −3.99935426808600744663568895732, −3.43300392687701213616157768787, −2.07751523981808632677167720922, −0.843500749294089367190579894856,
0.843500749294089367190579894856, 2.07751523981808632677167720922, 3.43300392687701213616157768787, 3.99935426808600744663568895732, 4.74640618715184076250855443997, 5.21116779224336242323026244517, 5.87742396321663981472484837893, 7.08596646204159274256067141042, 7.44503712049484423018361292011, 8.241359568143392443287980136853
