| L(s) = 1 | − 2.61·2-s + 4.85·4-s + 4.23·7-s − 7.47·8-s + 5.23·11-s + 3.23·13-s − 11.0·14-s + 9.85·16-s + 2.47·17-s − 1.76·19-s − 13.7·22-s + 3.23·23-s − 8.47·26-s + 20.5·28-s + 1.23·29-s − 31-s − 10.8·32-s − 6.47·34-s − 7.70·37-s + 4.61·38-s + 6.70·41-s − 3.70·43-s + 25.4·44-s − 8.47·46-s + 6.47·47-s + 10.9·49-s + 15.7·52-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.42·4-s + 1.60·7-s − 2.64·8-s + 1.57·11-s + 0.897·13-s − 2.96·14-s + 2.46·16-s + 0.599·17-s − 0.404·19-s − 2.92·22-s + 0.674·23-s − 1.66·26-s + 3.88·28-s + 0.229·29-s − 0.179·31-s − 1.91·32-s − 1.10·34-s − 1.26·37-s + 0.749·38-s + 1.04·41-s − 0.565·43-s + 3.83·44-s − 1.24·46-s + 0.944·47-s + 1.56·49-s + 2.17·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.404004769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404004769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 0.472T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
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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
−8.176943144797600692163316830511, −7.46678793643093042974345406712, −6.84083990367884074351175249459, −6.18651408576915534348282709819, −5.32374463752312124735591059500, −4.27018753402900427694999303957, −3.38353658455169727269024733178, −2.15331201020428823811795454662, −1.44319316653790295108272321235, −0.926517697221164114160201091296,
0.926517697221164114160201091296, 1.44319316653790295108272321235, 2.15331201020428823811795454662, 3.38353658455169727269024733178, 4.27018753402900427694999303957, 5.32374463752312124735591059500, 6.18651408576915534348282709819, 6.84083990367884074351175249459, 7.46678793643093042974345406712, 8.176943144797600692163316830511
