| L(s) = 1 | + 2·3-s + 26·7-s − 23·9-s + 28·11-s − 12·13-s + 64·17-s + 60·19-s + 52·21-s − 58·23-s − 100·27-s + 90·29-s + 128·31-s + 56·33-s − 236·37-s − 24·39-s + 242·41-s + 362·43-s + 226·47-s + 333·49-s + 128·51-s + 108·53-s + 120·57-s + 20·59-s + 542·61-s − 598·63-s − 434·67-s − 116·69-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.40·7-s − 0.851·9-s + 0.767·11-s − 0.256·13-s + 0.913·17-s + 0.724·19-s + 0.540·21-s − 0.525·23-s − 0.712·27-s + 0.576·29-s + 0.741·31-s + 0.295·33-s − 1.04·37-s − 0.0985·39-s + 0.921·41-s + 1.28·43-s + 0.701·47-s + 0.970·49-s + 0.351·51-s + 0.279·53-s + 0.278·57-s + 0.0441·59-s + 1.13·61-s − 1.19·63-s − 0.791·67-s − 0.202·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.577371748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.577371748\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 64 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 236 T + p^{3} T^{2} \) |
| 41 | \( 1 - 242 T + p^{3} T^{2} \) |
| 43 | \( 1 - 362 T + p^{3} T^{2} \) |
| 47 | \( 1 - 226 T + p^{3} T^{2} \) |
| 53 | \( 1 - 108 T + p^{3} T^{2} \) |
| 59 | \( 1 - 20 T + p^{3} T^{2} \) |
| 61 | \( 1 - 542 T + p^{3} T^{2} \) |
| 67 | \( 1 + 434 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 73 | \( 1 + 632 T + p^{3} T^{2} \) |
| 79 | \( 1 - 720 T + p^{3} T^{2} \) |
| 83 | \( 1 + 478 T + p^{3} T^{2} \) |
| 89 | \( 1 + 490 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1456 T + p^{3} T^{2} \) |
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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
−10.99097660295883712126484950532, −9.885114881853980922925741910869, −8.871895300321251516519372937078, −8.134223204023762689909108371238, −7.35271029384694112531935865199, −5.93812956316008483364736707181, −5.01952471074072067780625619037, −3.82163506616103598140046482124, −2.48144542929579332409402420826, −1.12119897084984035866273814000,
1.12119897084984035866273814000, 2.48144542929579332409402420826, 3.82163506616103598140046482124, 5.01952471074072067780625619037, 5.93812956316008483364736707181, 7.35271029384694112531935865199, 8.134223204023762689909108371238, 8.871895300321251516519372937078, 9.885114881853980922925741910869, 10.99097660295883712126484950532
