| L(s) = 1 | + 2-s + 2.86·3-s + 4-s − 0.300·5-s + 2.86·6-s + 3.26·7-s + 8-s + 5.18·9-s − 0.300·10-s − 2.00·11-s + 2.86·12-s − 1.10·13-s + 3.26·14-s − 0.859·15-s + 16-s − 4.69·17-s + 5.18·18-s − 7.03·19-s − 0.300·20-s + 9.35·21-s − 2.00·22-s − 5.22·23-s + 2.86·24-s − 4.90·25-s − 1.10·26-s + 6.26·27-s + 3.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.65·3-s + 0.5·4-s − 0.134·5-s + 1.16·6-s + 1.23·7-s + 0.353·8-s + 1.72·9-s − 0.0949·10-s − 0.603·11-s + 0.826·12-s − 0.306·13-s + 0.873·14-s − 0.221·15-s + 0.250·16-s − 1.13·17-s + 1.22·18-s − 1.61·19-s − 0.0671·20-s + 2.04·21-s − 0.426·22-s − 1.08·23-s + 0.584·24-s − 0.981·25-s − 0.216·26-s + 1.20·27-s + 0.617·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.036647014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.036647014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 5 | \( 1 + 0.300T + 5T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 + 2.00T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 + 4.69T + 17T^{2} \) |
| 19 | \( 1 + 7.03T + 19T^{2} \) |
| 23 | \( 1 + 5.22T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 9.69T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 7.01T + 53T^{2} \) |
| 59 | \( 1 - 9.07T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 5.35T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 5.80T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 8.72T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 - 3.98T + 97T^{2} \) |
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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
−10.38518303715768276907751318866, −9.167579237770437363992438132874, −8.397184699385745237543171098019, −7.85682105833796011098977876538, −7.04260668154901163164562368768, −5.72704797773411562295592898718, −4.39230473703123021467465733462, −4.03813000767411433414560955590, −2.46922009296965731758115460008, −2.04862781123983289905612657783,
2.04862781123983289905612657783, 2.46922009296965731758115460008, 4.03813000767411433414560955590, 4.39230473703123021467465733462, 5.72704797773411562295592898718, 7.04260668154901163164562368768, 7.85682105833796011098977876538, 8.397184699385745237543171098019, 9.167579237770437363992438132874, 10.38518303715768276907751318866
