| L(s) = 1 | − 0.600·3-s − 1.19·5-s − 0.0561·7-s − 2.63·9-s − 11-s − 0.130·13-s + 0.718·15-s − 0.761·17-s − 2.21·19-s + 0.0337·21-s + 5.28·23-s − 3.56·25-s + 3.38·27-s + 5.28·29-s − 3.33·31-s + 0.600·33-s + 0.0671·35-s + 4.35·37-s + 0.0781·39-s − 2.23·41-s − 6.81·43-s + 3.15·45-s − 11.8·47-s − 6.99·49-s + 0.457·51-s − 7.86·53-s + 1.19·55-s + ⋯ |
| L(s) = 1 | − 0.346·3-s − 0.535·5-s − 0.0212·7-s − 0.879·9-s − 0.301·11-s − 0.0360·13-s + 0.185·15-s − 0.184·17-s − 0.507·19-s + 0.00736·21-s + 1.10·23-s − 0.713·25-s + 0.651·27-s + 0.980·29-s − 0.598·31-s + 0.104·33-s + 0.0113·35-s + 0.715·37-s + 0.0125·39-s − 0.348·41-s − 1.03·43-s + 0.470·45-s − 1.73·47-s − 0.999·49-s + 0.0640·51-s − 1.08·53-s + 0.161·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8597055831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8597055831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + 0.600T + 3T^{2} \) |
| 5 | \( 1 + 1.19T + 5T^{2} \) |
| 7 | \( 1 + 0.0561T + 7T^{2} \) |
| 13 | \( 1 + 0.130T + 13T^{2} \) |
| 17 | \( 1 + 0.761T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 31 | \( 1 + 3.33T + 31T^{2} \) |
| 37 | \( 1 - 4.35T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 7.86T + 53T^{2} \) |
| 59 | \( 1 - 1.02T + 59T^{2} \) |
| 61 | \( 1 - 0.238T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 0.620T + 71T^{2} \) |
| 73 | \( 1 - 2.63T + 73T^{2} \) |
| 79 | \( 1 + 0.0287T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 7.53T + 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.249285140819882255217574509327, −7.37325920125045586119377429893, −6.57444290299196199776713363236, −6.04019604998199702302525969544, −5.06976894227150050525748424110, −4.65322810019286059208520104990, −3.51824334557793192368966298486, −2.93119205517153318221249772996, −1.84407881196418706431511080678, −0.47957913027633771320925087128,
0.47957913027633771320925087128, 1.84407881196418706431511080678, 2.93119205517153318221249772996, 3.51824334557793192368966298486, 4.65322810019286059208520104990, 5.06976894227150050525748424110, 6.04019604998199702302525969544, 6.57444290299196199776713363236, 7.37325920125045586119377429893, 8.249285140819882255217574509327
