L(s) = 1 | + 2-s − 1.08·3-s + 4-s − 2.98·5-s − 1.08·6-s − 1.13·7-s + 8-s − 1.83·9-s − 2.98·10-s − 2.91·11-s − 1.08·12-s − 0.731·13-s − 1.13·14-s + 3.22·15-s + 16-s + 7.72·17-s − 1.83·18-s − 2.28·19-s − 2.98·20-s + 1.22·21-s − 2.91·22-s + 0.277·23-s − 1.08·24-s + 3.89·25-s − 0.731·26-s + 5.22·27-s − 1.13·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.624·3-s + 0.5·4-s − 1.33·5-s − 0.441·6-s − 0.428·7-s + 0.353·8-s − 0.610·9-s − 0.943·10-s − 0.878·11-s − 0.312·12-s − 0.202·13-s − 0.303·14-s + 0.832·15-s + 0.250·16-s + 1.87·17-s − 0.431·18-s − 0.524·19-s − 0.666·20-s + 0.267·21-s − 0.621·22-s + 0.0579·23-s − 0.220·24-s + 0.779·25-s − 0.143·26-s + 1.00·27-s − 0.214·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 0.731T + 13T^{2} \) |
| 17 | \( 1 - 7.72T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 - 0.277T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 - 7.72T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 + 2.69T + 41T^{2} \) |
| 43 | \( 1 - 1.83T + 43T^{2} \) |
| 47 | \( 1 + 2.34T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 4.81T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 6.42T + 67T^{2} \) |
| 71 | \( 1 + 8.65T + 71T^{2} \) |
| 73 | \( 1 + 2.20T + 73T^{2} \) |
| 79 | \( 1 - 7.73T + 79T^{2} \) |
| 83 | \( 1 - 5.72T + 83T^{2} \) |
| 89 | \( 1 + 7.04T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.52384712908770622806452935135, −6.63120849042627302100228010280, −6.05793613539293694284333306014, −5.28740633064618834503479229059, −4.75270195577761164308710870603, −3.94768012997956384871513604585, −3.10788686152781151686138534149, −2.70013233804403977088496447994, −1.02942671504246984948162929888, 0,
1.02942671504246984948162929888, 2.70013233804403977088496447994, 3.10788686152781151686138534149, 3.94768012997956384871513604585, 4.75270195577761164308710870603, 5.28740633064618834503479229059, 6.05793613539293694284333306014, 6.63120849042627302100228010280, 7.52384712908770622806452935135