| L(s) = 1 | + 0.417·2-s − 3-s − 1.82·4-s + 1.29·5-s − 0.417·6-s − 7-s − 1.59·8-s + 9-s + 0.541·10-s + 4.81·11-s + 1.82·12-s + 4.21·13-s − 0.417·14-s − 1.29·15-s + 2.98·16-s − 6.66·17-s + 0.417·18-s + 2.36·19-s − 2.36·20-s + 21-s + 2.00·22-s − 2.22·23-s + 1.59·24-s − 3.31·25-s + 1.75·26-s − 27-s + 1.82·28-s + ⋯ |
| L(s) = 1 | + 0.295·2-s − 0.577·3-s − 0.912·4-s + 0.579·5-s − 0.170·6-s − 0.377·7-s − 0.564·8-s + 0.333·9-s + 0.171·10-s + 1.45·11-s + 0.527·12-s + 1.16·13-s − 0.111·14-s − 0.334·15-s + 0.746·16-s − 1.61·17-s + 0.0983·18-s + 0.542·19-s − 0.529·20-s + 0.218·21-s + 0.428·22-s − 0.463·23-s + 0.325·24-s − 0.663·25-s + 0.344·26-s − 0.192·27-s + 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 0.417T + 2T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 11 | \( 1 - 4.81T + 11T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 17 | \( 1 + 6.66T + 17T^{2} \) |
| 19 | \( 1 - 2.36T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 7.87T + 41T^{2} \) |
| 43 | \( 1 + 7.26T + 43T^{2} \) |
| 47 | \( 1 + 3.03T + 47T^{2} \) |
| 53 | \( 1 - 7.86T + 53T^{2} \) |
| 59 | \( 1 + 7.45T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 + 3.42T + 79T^{2} \) |
| 83 | \( 1 - 0.221T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 7.93T + 97T^{2} \) |
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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
−7.28125206552510130373819689198, −6.51147631385857623232460628161, −6.08949835528894859471831356643, −5.48494297023561148192842013894, −4.64192364398573585551449781529, −3.86597835159539062239898708466, −3.53628673658403502600782032290, −2.06886461564622422883433348715, −1.19804552112824130948259296200, 0,
1.19804552112824130948259296200, 2.06886461564622422883433348715, 3.53628673658403502600782032290, 3.86597835159539062239898708466, 4.64192364398573585551449781529, 5.48494297023561148192842013894, 6.08949835528894859471831356643, 6.51147631385857623232460628161, 7.28125206552510130373819689198
