| L(s) = 1 | − 0.785·2-s − 7.38·4-s − 20.9·7-s + 12.0·8-s + 59.3·11-s + 45.9·13-s + 16.4·14-s + 49.5·16-s + 43.0·17-s + 140.·19-s − 46.6·22-s − 101.·23-s − 36.0·26-s + 154.·28-s − 12.3·29-s + 71.9·31-s − 135.·32-s − 33.8·34-s + 150.·37-s − 110.·38-s − 51.0·41-s + 34.4·43-s − 438.·44-s + 79.9·46-s + 117.·47-s + 94.3·49-s − 339.·52-s + ⋯ |
| L(s) = 1 | − 0.277·2-s − 0.922·4-s − 1.12·7-s + 0.533·8-s + 1.62·11-s + 0.979·13-s + 0.313·14-s + 0.774·16-s + 0.614·17-s + 1.69·19-s − 0.451·22-s − 0.922·23-s − 0.272·26-s + 1.04·28-s − 0.0793·29-s + 0.417·31-s − 0.748·32-s − 0.170·34-s + 0.669·37-s − 0.469·38-s − 0.194·41-s + 0.122·43-s − 1.50·44-s + 0.256·46-s + 0.366·47-s + 0.275·49-s − 0.904·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.594933691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.594933691\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.785T + 8T^{2} \) |
| 7 | \( 1 + 20.9T + 343T^{2} \) |
| 11 | \( 1 - 59.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 12.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 71.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 51.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 34.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 496.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 247.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 826.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 260.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 372.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 476.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 313.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 817.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 941.T + 9.12e5T^{2} \) |
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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
−9.017416973796268208103365026108, −8.116618748387158802007389131553, −7.31257096057031961760269099139, −6.28679332321940045092959095927, −5.81159314846956382558638640206, −4.61378767720681417802150115782, −3.69359109727718612443013503498, −3.25939487167720599774448283593, −1.44568713540376133810607024517, −0.68289884495884877015638892587,
0.68289884495884877015638892587, 1.44568713540376133810607024517, 3.25939487167720599774448283593, 3.69359109727718612443013503498, 4.61378767720681417802150115782, 5.81159314846956382558638640206, 6.28679332321940045092959095927, 7.31257096057031961760269099139, 8.116618748387158802007389131553, 9.017416973796268208103365026108
