| L(s) = 1 | + 2-s + 2.60·3-s + 4-s − 2.07·5-s + 2.60·6-s + 4.23·7-s + 8-s + 3.78·9-s − 2.07·10-s + 3.41·11-s + 2.60·12-s + 4.28·13-s + 4.23·14-s − 5.40·15-s + 16-s − 4.65·17-s + 3.78·18-s + 6.08·19-s − 2.07·20-s + 11.0·21-s + 3.41·22-s − 23-s + 2.60·24-s − 0.698·25-s + 4.28·26-s + 2.05·27-s + 4.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.50·3-s + 0.5·4-s − 0.927·5-s + 1.06·6-s + 1.59·7-s + 0.353·8-s + 1.26·9-s − 0.655·10-s + 1.02·11-s + 0.752·12-s + 1.18·13-s + 1.13·14-s − 1.39·15-s + 0.250·16-s − 1.12·17-s + 0.893·18-s + 1.39·19-s − 0.463·20-s + 2.40·21-s + 0.727·22-s − 0.208·23-s + 0.531·24-s − 0.139·25-s + 0.840·26-s + 0.395·27-s + 0.799·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.538955844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.538955844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 + 4.65T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 29 | \( 1 - 0.279T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 + 0.135T + 41T^{2} \) |
| 43 | \( 1 + 8.49T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.74T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 9.69T + 61T^{2} \) |
| 67 | \( 1 - 8.83T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 9.49T + 73T^{2} \) |
| 79 | \( 1 + 7.09T + 79T^{2} \) |
| 83 | \( 1 + 2.67T + 83T^{2} \) |
| 89 | \( 1 + 2.92T + 89T^{2} \) |
| 97 | \( 1 + 8.39T + 97T^{2} \) |
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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
−7.997719966214625937165102326950, −7.66144313169375635513622342073, −6.79569987554147349222191507217, −5.95882804778808504570886263550, −4.74352610745639843233569611971, −4.37760568024807595079425627463, −3.60005105151495284884680930616, −3.08437357471804584482798856538, −1.89015493086013866952229445424, −1.32893916827268820189486499192,
1.32893916827268820189486499192, 1.89015493086013866952229445424, 3.08437357471804584482798856538, 3.60005105151495284884680930616, 4.37760568024807595079425627463, 4.74352610745639843233569611971, 5.95882804778808504570886263550, 6.79569987554147349222191507217, 7.66144313169375635513622342073, 7.997719966214625937165102326950
