| L(s) = 1 | + 2.07·2-s − 3.70·4-s + 5·5-s − 4.66·7-s − 24.2·8-s + 10.3·10-s + 8.89·11-s + 34.6·13-s − 9.66·14-s − 20.6·16-s + 2.66·17-s + 125.·19-s − 18.5·20-s + 18.4·22-s + 131.·23-s + 25·25-s + 71.7·26-s + 17.2·28-s − 71.2·29-s + 13.4·31-s + 151.·32-s + 5.52·34-s − 23.3·35-s + 283.·37-s + 260.·38-s − 121.·40-s + 383.·41-s + ⋯ |
| L(s) = 1 | + 0.733·2-s − 0.462·4-s + 0.447·5-s − 0.251·7-s − 1.07·8-s + 0.327·10-s + 0.243·11-s + 0.738·13-s − 0.184·14-s − 0.323·16-s + 0.0380·17-s + 1.51·19-s − 0.206·20-s + 0.178·22-s + 1.19·23-s + 0.200·25-s + 0.541·26-s + 0.116·28-s − 0.455·29-s + 0.0777·31-s + 0.835·32-s + 0.0278·34-s − 0.112·35-s + 1.26·37-s + 1.11·38-s − 0.479·40-s + 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.550759914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550759914\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 2.07T + 8T^{2} \) |
| 7 | \( 1 + 4.66T + 343T^{2} \) |
| 11 | \( 1 - 8.89T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.66T + 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 71.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 13.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 383.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 339.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 78.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 254.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 32.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 186.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 406.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 966.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 276.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 178.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 806.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−11.00730962300475955537977800442, −9.677563658832226758726870914296, −9.237315340018856220436448359424, −8.124856283909253215993979305837, −6.82251032880966103054156152019, −5.82818743465128483532909098281, −5.05155145804377602532317518701, −3.85030636186845780042917341616, −2.87825579547712393997145342058, −0.987160427829812003179293441852,
0.987160427829812003179293441852, 2.87825579547712393997145342058, 3.85030636186845780042917341616, 5.05155145804377602532317518701, 5.82818743465128483532909098281, 6.82251032880966103054156152019, 8.124856283909253215993979305837, 9.237315340018856220436448359424, 9.677563658832226758726870914296, 11.00730962300475955537977800442
