| L(s) = 1 | + 0.242·3-s − 0.379·5-s − 7-s − 2.94·9-s − 2.61·11-s + 2.31·13-s − 0.0919·15-s + 7.37·17-s − 5.44·19-s − 0.242·21-s − 6.44·23-s − 4.85·25-s − 1.43·27-s − 5.07·29-s − 6.10·31-s − 0.633·33-s + 0.379·35-s + 10.4·37-s + 0.560·39-s − 0.836·41-s + 5.50·43-s + 1.11·45-s + 6.02·47-s + 49-s + 1.78·51-s − 0.813·53-s + 0.992·55-s + ⋯ |
| L(s) = 1 | + 0.139·3-s − 0.169·5-s − 0.377·7-s − 0.980·9-s − 0.788·11-s + 0.641·13-s − 0.0237·15-s + 1.78·17-s − 1.24·19-s − 0.0528·21-s − 1.34·23-s − 0.971·25-s − 0.276·27-s − 0.941·29-s − 1.09·31-s − 0.110·33-s + 0.0641·35-s + 1.72·37-s + 0.0896·39-s − 0.130·41-s + 0.838·43-s + 0.166·45-s + 0.878·47-s + 0.142·49-s + 0.249·51-s − 0.111·53-s + 0.133·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.204279603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204279603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 0.242T + 3T^{2} \) |
| 5 | \( 1 + 0.379T + 5T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 + 6.10T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 0.836T + 41T^{2} \) |
| 43 | \( 1 - 5.50T + 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 53 | \( 1 + 0.813T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 + 1.31T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 - 4.21T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 - 9.32T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.889353729109613707178982640087, −7.51888680059316864854932536920, −6.32620925943142810442036138466, −5.77280977304246152494968388102, −5.40310381286670707855557963856, −4.06471165362089699749550806260, −3.66137903282831269878199891412, −2.70620620956903716992963900722, −1.95766369362148526571275927264, −0.52655069349625856972413198448,
0.52655069349625856972413198448, 1.95766369362148526571275927264, 2.70620620956903716992963900722, 3.66137903282831269878199891412, 4.06471165362089699749550806260, 5.40310381286670707855557963856, 5.77280977304246152494968388102, 6.32620925943142810442036138466, 7.51888680059316864854932536920, 7.889353729109613707178982640087
