| L(s) = 1 | − 0.199·2-s − 3.23·3-s − 1.96·4-s + 5-s + 0.645·6-s − 5.08·7-s + 0.790·8-s + 7.45·9-s − 0.199·10-s + 11-s + 6.33·12-s + 1.81·13-s + 1.01·14-s − 3.23·15-s + 3.76·16-s + 2.44·17-s − 1.48·18-s − 3.81·19-s − 1.96·20-s + 16.4·21-s − 0.199·22-s + 1.59·23-s − 2.55·24-s + 25-s − 0.361·26-s − 14.4·27-s + 9.97·28-s + ⋯ |
| L(s) = 1 | − 0.141·2-s − 1.86·3-s − 0.980·4-s + 0.447·5-s + 0.263·6-s − 1.92·7-s + 0.279·8-s + 2.48·9-s − 0.0630·10-s + 0.301·11-s + 1.82·12-s + 0.503·13-s + 0.271·14-s − 0.834·15-s + 0.940·16-s + 0.593·17-s − 0.350·18-s − 0.874·19-s − 0.438·20-s + 3.59·21-s − 0.0425·22-s + 0.333·23-s − 0.521·24-s + 0.200·25-s − 0.0709·26-s − 2.77·27-s + 1.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3030561384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3030561384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 0.199T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 13 | \( 1 - 1.81T + 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 + 3.81T + 19T^{2} \) |
| 23 | \( 1 - 1.59T + 23T^{2} \) |
| 29 | \( 1 + 0.667T + 29T^{2} \) |
| 31 | \( 1 + 0.936T + 31T^{2} \) |
| 37 | \( 1 + 9.86T + 37T^{2} \) |
| 41 | \( 1 - 1.75T + 41T^{2} \) |
| 43 | \( 1 + 1.98T + 43T^{2} \) |
| 47 | \( 1 - 8.88T + 47T^{2} \) |
| 53 | \( 1 + 7.64T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 4.97T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 7.86T + 71T^{2} \) |
| 79 | \( 1 + 3.69T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 1.50T + 89T^{2} \) |
| 97 | \( 1 + 2.22T + 97T^{2} \) |
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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
−8.727428402036933537761864303574, −7.39791024633333440968324741869, −6.74222321087496118800926440606, −6.00907758237977674879760248983, −5.76349066970330007407948139650, −4.81050355853050032541605505138, −4.02156418218475670499868475413, −3.23360500208082317986724390989, −1.45188416573991716864206229714, −0.38085981519290834752066702583,
0.38085981519290834752066702583, 1.45188416573991716864206229714, 3.23360500208082317986724390989, 4.02156418218475670499868475413, 4.81050355853050032541605505138, 5.76349066970330007407948139650, 6.00907758237977674879760248983, 6.74222321087496118800926440606, 7.39791024633333440968324741869, 8.727428402036933537761864303574
