L(s) = 1 | − 1.82·2-s − 2.88·3-s + 1.32·4-s − 5-s + 5.26·6-s + 5.06·7-s + 1.23·8-s + 5.34·9-s + 1.82·10-s + 11-s − 3.81·12-s + 6.98·13-s − 9.23·14-s + 2.88·15-s − 4.89·16-s − 17-s − 9.73·18-s − 2.97·19-s − 1.32·20-s − 14.6·21-s − 1.82·22-s + 4.46·23-s − 3.57·24-s + 25-s − 12.7·26-s − 6.77·27-s + 6.68·28-s + ⋯ |
L(s) = 1 | − 1.28·2-s − 1.66·3-s + 0.660·4-s − 0.447·5-s + 2.14·6-s + 1.91·7-s + 0.437·8-s + 1.78·9-s + 0.576·10-s + 0.301·11-s − 1.10·12-s + 1.93·13-s − 2.46·14-s + 0.745·15-s − 1.22·16-s − 0.242·17-s − 2.29·18-s − 0.681·19-s − 0.295·20-s − 3.19·21-s − 0.388·22-s + 0.931·23-s − 0.730·24-s + 0.200·25-s − 2.49·26-s − 1.30·27-s + 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5728958282\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5728958282\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 + 2.88T + 3T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 13 | \( 1 - 6.98T + 13T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 + 7.16T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 + 6.87T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 7.62T + 59T^{2} \) |
| 61 | \( 1 + 9.78T + 61T^{2} \) |
| 67 | \( 1 - 1.45T + 67T^{2} \) |
| 71 | \( 1 + 8.24T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 - 4.77T + 83T^{2} \) |
| 89 | \( 1 + 2.28T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−10.49139533776085740118009685819, −9.066408140180800421109532330127, −8.414560268203727546729907363043, −7.70628497111487832254274043897, −6.76023196878638014962852517153, −5.85067447034482440107302733339, −4.78473586760579322510650224966, −4.18298204061930412178615400683, −1.61954355892429300149154306138, −0.890665957798294193287741453833,
0.890665957798294193287741453833, 1.61954355892429300149154306138, 4.18298204061930412178615400683, 4.78473586760579322510650224966, 5.85067447034482440107302733339, 6.76023196878638014962852517153, 7.70628497111487832254274043897, 8.414560268203727546729907363043, 9.066408140180800421109532330127, 10.49139533776085740118009685819