| L(s) = 1 | + 0.480·2-s + 1.60·3-s − 1.76·4-s + 0.770·6-s − 0.480·7-s − 1.81·8-s − 0.434·9-s − 2.83·12-s + 4.79·13-s − 0.231·14-s + 2.66·16-s − 2.50·17-s − 0.209·18-s − 5.75·19-s − 0.770·21-s − 4.43·23-s − 2.90·24-s + 2.30·26-s − 5.50·27-s + 0.850·28-s + 9.01·29-s + 7.97·31-s + 4.90·32-s − 1.20·34-s + 0.768·36-s − 5.20·37-s − 2.76·38-s + ⋯ |
| L(s) = 1 | + 0.340·2-s + 0.924·3-s − 0.884·4-s + 0.314·6-s − 0.181·7-s − 0.640·8-s − 0.144·9-s − 0.817·12-s + 1.33·13-s − 0.0618·14-s + 0.666·16-s − 0.606·17-s − 0.0492·18-s − 1.32·19-s − 0.168·21-s − 0.924·23-s − 0.592·24-s + 0.452·26-s − 1.05·27-s + 0.160·28-s + 1.67·29-s + 1.43·31-s + 0.867·32-s − 0.206·34-s + 0.128·36-s − 0.855·37-s − 0.449·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.480T + 2T^{2} \) |
| 3 | \( 1 - 1.60T + 3T^{2} \) |
| 7 | \( 1 + 0.480T + 7T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 + 2.50T + 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 - 9.01T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 + 8.45T + 41T^{2} \) |
| 43 | \( 1 + 8.53T + 43T^{2} \) |
| 47 | \( 1 + 9.60T + 47T^{2} \) |
| 53 | \( 1 + 6.10T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 + 2.76T + 71T^{2} \) |
| 73 | \( 1 - 5.00T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 - 1.71T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 3.63T + 97T^{2} \) |
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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
−8.555496570972709852015192776348, −8.081292106443703689075683882567, −6.52149068031639922601398291362, −6.26068506996697352709676751207, −5.06188606453771335695397905724, −4.30952004586541261183026554534, −3.54053656229480300384368295044, −2.87234895041246934542477708720, −1.64705113091962688748788486401, 0,
1.64705113091962688748788486401, 2.87234895041246934542477708720, 3.54053656229480300384368295044, 4.30952004586541261183026554534, 5.06188606453771335695397905724, 6.26068506996697352709676751207, 6.52149068031639922601398291362, 8.081292106443703689075683882567, 8.555496570972709852015192776348
