| L(s) = 1 | + 2.79·2-s + 6.21·3-s − 0.176·4-s + 17.3·6-s − 22.8·8-s + 11.6·9-s − 49.6·11-s − 1.09·12-s + 71.6·13-s − 62.5·16-s + 46.2·17-s + 32.4·18-s − 54.1·19-s − 138.·22-s − 152.·23-s − 142.·24-s + 200.·26-s − 95.6·27-s − 38.5·29-s − 103.·31-s + 7.98·32-s − 308.·33-s + 129.·34-s − 2.04·36-s − 53.4·37-s − 151.·38-s + 445.·39-s + ⋯ |
| L(s) = 1 | + 0.988·2-s + 1.19·3-s − 0.0220·4-s + 1.18·6-s − 1.01·8-s + 0.430·9-s − 1.35·11-s − 0.0263·12-s + 1.52·13-s − 0.977·16-s + 0.660·17-s + 0.425·18-s − 0.654·19-s − 1.34·22-s − 1.38·23-s − 1.20·24-s + 1.51·26-s − 0.681·27-s − 0.246·29-s − 0.599·31-s + 0.0441·32-s − 1.62·33-s + 0.653·34-s − 0.00948·36-s − 0.237·37-s − 0.647·38-s + 1.82·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.79T + 8T^{2} \) |
| 3 | \( 1 - 6.21T + 27T^{2} \) |
| 11 | \( 1 + 49.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 71.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 38.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 103.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 53.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 40.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 39.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 197.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 972.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 386.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 219.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 386.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 750.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 533.T + 9.12e5T^{2} \) |
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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
−8.647970765311627725182281141859, −8.346914018111439249049274642878, −7.44442148377961494778980410499, −6.08530608082975770056613838788, −5.56934854492299896133346017012, −4.39582266803378934029417358137, −3.57831599753960780119189305674, −2.95679975816555638094237878528, −1.87073614740930124022956926751, 0,
1.87073614740930124022956926751, 2.95679975816555638094237878528, 3.57831599753960780119189305674, 4.39582266803378934029417358137, 5.56934854492299896133346017012, 6.08530608082975770056613838788, 7.44442148377961494778980410499, 8.346914018111439249049274642878, 8.647970765311627725182281141859
