| L(s) = 1 | − 2-s + 1.76·3-s + 4-s − 2.10·5-s − 1.76·6-s + 4.17·7-s − 8-s + 0.123·9-s + 2.10·10-s − 3.67·11-s + 1.76·12-s + 3.27·13-s − 4.17·14-s − 3.71·15-s + 16-s + 6.59·17-s − 0.123·18-s + 7.83·19-s − 2.10·20-s + 7.37·21-s + 3.67·22-s − 23-s − 1.76·24-s − 0.573·25-s − 3.27·26-s − 5.08·27-s + 4.17·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.02·3-s + 0.5·4-s − 0.940·5-s − 0.721·6-s + 1.57·7-s − 0.353·8-s + 0.0410·9-s + 0.665·10-s − 1.10·11-s + 0.510·12-s + 0.908·13-s − 1.11·14-s − 0.960·15-s + 0.250·16-s + 1.60·17-s − 0.0290·18-s + 1.79·19-s − 0.470·20-s + 1.60·21-s + 0.784·22-s − 0.208·23-s − 0.360·24-s − 0.114·25-s − 0.642·26-s − 0.978·27-s + 0.788·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.229940353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.229940353\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 2.10T + 5T^{2} \) |
| 7 | \( 1 - 4.17T + 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 13 | \( 1 - 3.27T + 13T^{2} \) |
| 17 | \( 1 - 6.59T + 17T^{2} \) |
| 19 | \( 1 - 7.83T + 19T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 7.30T + 37T^{2} \) |
| 41 | \( 1 + 0.848T + 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 + 0.139T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 - 8.52T + 59T^{2} \) |
| 61 | \( 1 + 7.88T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 2.18T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 9.84T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 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
−8.027392016366875882794732900587, −7.86745880453707466679064467993, −7.21520749855951523712110564213, −5.83272380712332734954220846066, −5.25948637995905426604009114795, −4.32284431729150423192886452629, −3.32759199941961717800704953466, −2.88773132440955161490199128572, −1.72579662864258159013056298248, −0.880176545516630599223971836746,
0.880176545516630599223971836746, 1.72579662864258159013056298248, 2.88773132440955161490199128572, 3.32759199941961717800704953466, 4.32284431729150423192886452629, 5.25948637995905426604009114795, 5.83272380712332734954220846066, 7.21520749855951523712110564213, 7.86745880453707466679064467993, 8.027392016366875882794732900587
