L(s) = 1 | + 2-s − 3-s + 4-s + 3.88·5-s − 6-s − 0.785·7-s + 8-s + 9-s + 3.88·10-s + 4.53·11-s − 12-s + 6.52·13-s − 0.785·14-s − 3.88·15-s + 16-s + 2·17-s + 18-s − 19-s + 3.88·20-s + 0.785·21-s + 4.53·22-s − 1.34·23-s − 24-s + 10.1·25-s + 6.52·26-s − 27-s − 0.785·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.73·5-s − 0.408·6-s − 0.296·7-s + 0.353·8-s + 0.333·9-s + 1.22·10-s + 1.36·11-s − 0.288·12-s + 1.81·13-s − 0.209·14-s − 1.00·15-s + 0.250·16-s + 0.485·17-s + 0.235·18-s − 0.229·19-s + 0.869·20-s + 0.171·21-s + 0.967·22-s − 0.281·23-s − 0.204·24-s + 2.02·25-s + 1.28·26-s − 0.192·27-s − 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.636004324\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.636004324\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 + 0.785T + 7T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 - 6.52T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 1.34T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 - 0.236T + 31T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 + 2.65T + 47T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 + 1.81T + 61T^{2} \) |
| 67 | \( 1 + 2.50T + 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 - 3.73T + 73T^{2} \) |
| 79 | \( 1 + 7.44T + 79T^{2} \) |
| 83 | \( 1 - 3.60T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−8.099518658628594779646409689546, −6.73952406757450715577763383965, −6.50799224957048613488400496766, −5.95622904390597429121355767089, −5.45585123407634888235413760433, −4.50145100415842077449015227808, −3.71116122281191047627119257837, −2.87214215823742210825171056501, −1.62674994388021305431788481770, −1.25783968468931333860484642273,
1.25783968468931333860484642273, 1.62674994388021305431788481770, 2.87214215823742210825171056501, 3.71116122281191047627119257837, 4.50145100415842077449015227808, 5.45585123407634888235413760433, 5.95622904390597429121355767089, 6.50799224957048613488400496766, 6.73952406757450715577763383965, 8.099518658628594779646409689546