| L(s) = 1 | − 3.56e5·3-s − 8.06e7·5-s − 9.58e9·7-s + 3.30e10·9-s + 1.65e12·11-s + 3.34e12·13-s + 2.87e13·15-s + 5.54e13·17-s − 1.98e14·19-s + 3.41e15·21-s + 2.67e15·23-s − 5.42e15·25-s + 2.17e16·27-s + 1.00e16·29-s − 2.05e17·31-s − 5.88e17·33-s + 7.72e17·35-s − 1.65e18·37-s − 1.19e18·39-s − 6.38e18·41-s − 6.62e18·43-s − 2.66e18·45-s − 3.90e17·47-s + 6.45e19·49-s − 1.97e19·51-s − 5.42e19·53-s − 1.33e20·55-s + ⋯ |
| L(s) = 1 | − 1.16·3-s − 0.738·5-s − 1.83·7-s + 0.350·9-s + 1.74·11-s + 0.517·13-s + 0.858·15-s + 0.392·17-s − 0.390·19-s + 2.12·21-s + 0.585·23-s − 0.454·25-s + 0.754·27-s + 0.153·29-s − 1.45·31-s − 2.02·33-s + 1.35·35-s − 1.53·37-s − 0.601·39-s − 1.81·41-s − 1.08·43-s − 0.258·45-s − 0.0230·47-s + 2.35·49-s − 0.456·51-s − 0.803·53-s − 1.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.2716993085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2716993085\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 3.56e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 8.06e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 9.58e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.65e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 3.34e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 5.54e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 1.98e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 2.67e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.00e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 2.05e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.65e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 6.38e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 6.62e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 3.90e17T + 2.87e38T^{2} \) |
| 53 | \( 1 + 5.42e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.72e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 1.88e19T + 1.15e41T^{2} \) |
| 67 | \( 1 - 9.12e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 3.13e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 4.37e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.12e22T + 4.42e43T^{2} \) |
| 83 | \( 1 - 3.90e21T + 1.37e44T^{2} \) |
| 89 | \( 1 + 7.96e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 6.63e22T + 4.96e45T^{2} \) |
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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
−10.81181992851765531649536286387, −9.674519520386724264725983498741, −8.682170489790027708836866654553, −6.86080390629950268218270792987, −6.50607317972899037274050044324, −5.40605739429222239891178190278, −3.87599280350020312742372930552, −3.33813305257518831852878629032, −1.40592160734461248663000867969, −0.24298663109883912291533648127,
0.24298663109883912291533648127, 1.40592160734461248663000867969, 3.33813305257518831852878629032, 3.87599280350020312742372930552, 5.40605739429222239891178190278, 6.50607317972899037274050044324, 6.86080390629950268218270792987, 8.682170489790027708836866654553, 9.674519520386724264725983498741, 10.81181992851765531649536286387
