L(s) = 1 | + 2.01·2-s − 3.02·3-s + 2.05·4-s − 6.09·6-s − 0.369·7-s + 0.110·8-s + 6.15·9-s − 1.74·11-s − 6.21·12-s − 1.11·13-s − 0.745·14-s − 3.88·16-s − 5.48·17-s + 12.3·18-s − 3.75·19-s + 1.11·21-s − 3.51·22-s − 7.24·23-s − 0.334·24-s − 2.24·26-s − 9.54·27-s − 0.760·28-s + 4.19·29-s + 0.305·31-s − 8.04·32-s + 5.28·33-s − 11.0·34-s + ⋯ |
L(s) = 1 | + 1.42·2-s − 1.74·3-s + 1.02·4-s − 2.48·6-s − 0.139·7-s + 0.0390·8-s + 2.05·9-s − 0.526·11-s − 1.79·12-s − 0.309·13-s − 0.199·14-s − 0.971·16-s − 1.33·17-s + 2.92·18-s − 0.860·19-s + 0.244·21-s − 0.749·22-s − 1.51·23-s − 0.0682·24-s − 0.440·26-s − 1.83·27-s − 0.143·28-s + 0.778·29-s + 0.0549·31-s − 1.42·32-s + 0.919·33-s − 1.89·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 2 | \( 1 - 2.01T + 2T^{2} \) |
| 3 | \( 1 + 3.02T + 3T^{2} \) |
| 7 | \( 1 + 0.369T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 + 3.75T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 - 0.305T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 + 7.17T + 43T^{2} \) |
| 47 | \( 1 + 0.810T + 47T^{2} \) |
| 53 | \( 1 - 3.91T + 53T^{2} \) |
| 59 | \( 1 + 1.85T + 59T^{2} \) |
| 61 | \( 1 - 9.68T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 3.43T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 + 7.13T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−10.56356446418444198108504227153, −9.709106585352312668339432853073, −8.209179149340250429035618455472, −6.71230479282911145958149642670, −6.40057746248053995298249954884, −5.44868247041306516499271923932, −4.71459461710063960382776299242, −4.00151690301899430342362347010, −2.30024905987496315178555155490, 0,
2.30024905987496315178555155490, 4.00151690301899430342362347010, 4.71459461710063960382776299242, 5.44868247041306516499271923932, 6.40057746248053995298249954884, 6.71230479282911145958149642670, 8.209179149340250429035618455472, 9.709106585352312668339432853073, 10.56356446418444198108504227153