L(s) = 1 | − 2-s − 3-s + 4-s − 0.757·5-s + 6-s − 3.37·7-s − 8-s + 9-s + 0.757·10-s + 11-s − 12-s − 4.33·13-s + 3.37·14-s + 0.757·15-s + 16-s − 7.25·17-s − 18-s − 7.88·19-s − 0.757·20-s + 3.37·21-s − 22-s − 6.47·23-s + 24-s − 4.42·25-s + 4.33·26-s − 27-s − 3.37·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.338·5-s + 0.408·6-s − 1.27·7-s − 0.353·8-s + 0.333·9-s + 0.239·10-s + 0.301·11-s − 0.288·12-s − 1.20·13-s + 0.901·14-s + 0.195·15-s + 0.250·16-s − 1.76·17-s − 0.235·18-s − 1.80·19-s − 0.169·20-s + 0.735·21-s − 0.213·22-s − 1.35·23-s + 0.204·24-s − 0.885·25-s + 0.850·26-s − 0.192·27-s − 0.637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06398105739\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06398105739\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 + 0.757T + 5T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 + 7.25T + 17T^{2} \) |
| 19 | \( 1 + 7.88T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 6.28T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 - 4.06T + 41T^{2} \) |
| 43 | \( 1 + 0.645T + 43T^{2} \) |
| 47 | \( 1 + 5.39T + 47T^{2} \) |
| 53 | \( 1 - 6.50T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 5.41T + 73T^{2} \) |
| 79 | \( 1 + 9.81T + 79T^{2} \) |
| 83 | \( 1 + 2.63T + 83T^{2} \) |
| 89 | \( 1 - 6.54T + 89T^{2} \) |
| 97 | \( 1 - 0.201T + 97T^{2} \) |
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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
−8.537913412199715050707641744836, −7.67522251482301513196017168772, −6.88238131996730060919300470041, −6.38801336520380076846742121143, −5.83166025431330144372108539209, −4.44773354227411679888333954254, −4.03919855701810014979898449342, −2.68529044138012132588155103306, −1.97580597138797820330126330594, −0.15288877524271818723834015230,
0.15288877524271818723834015230, 1.97580597138797820330126330594, 2.68529044138012132588155103306, 4.03919855701810014979898449342, 4.44773354227411679888333954254, 5.83166025431330144372108539209, 6.38801336520380076846742121143, 6.88238131996730060919300470041, 7.67522251482301513196017168772, 8.537913412199715050707641744836