| L(s) = 1 | − 2.66·2-s + 5.12·4-s − 1.45·5-s − 8.33·8-s + 3.88·10-s + 1.54·11-s − 5.88·13-s + 12.0·16-s − 6.79·17-s + 6.24·19-s − 7.45·20-s − 4.12·22-s − 2.90·23-s − 2.88·25-s + 15.7·26-s − 3.88·29-s − 2·31-s − 15.3·32-s + 18.1·34-s + 5·37-s − 16.6·38-s + 12.1·40-s − 2.24·41-s − 7.13·43-s + 7.91·44-s + 7.76·46-s − 5.33·47-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.56·4-s − 0.650·5-s − 2.94·8-s + 1.22·10-s + 0.465·11-s − 1.63·13-s + 3.00·16-s − 1.64·17-s + 1.43·19-s − 1.66·20-s − 0.879·22-s − 0.606·23-s − 0.576·25-s + 3.07·26-s − 0.721·29-s − 0.359·31-s − 2.71·32-s + 3.10·34-s + 0.821·37-s − 2.70·38-s + 1.91·40-s − 0.351·41-s − 1.08·43-s + 1.19·44-s + 1.14·46-s − 0.778·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3147219867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3147219867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 + 5.88T + 13T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 + 7.13T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 + 4.67T + 59T^{2} \) |
| 61 | \( 1 - 2.36T + 61T^{2} \) |
| 67 | \( 1 - 3.36T + 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 - 1.88T + 73T^{2} \) |
| 79 | \( 1 - 3.36T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 0.793T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.428604618766981059039194430850, −7.83652009135025468880415632865, −7.20119612660951083291530085392, −6.79120366243504666352524856966, −5.78064480065519281009912952176, −4.69862633264689876250437642122, −3.57185673553874596429523635695, −2.51348762964771214451820889214, −1.77554554952848420857948929480, −0.41081771851745334484665875597,
0.41081771851745334484665875597, 1.77554554952848420857948929480, 2.51348762964771214451820889214, 3.57185673553874596429523635695, 4.69862633264689876250437642122, 5.78064480065519281009912952176, 6.79120366243504666352524856966, 7.20119612660951083291530085392, 7.83652009135025468880415632865, 8.428604618766981059039194430850
