| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s − 2·11-s + 13-s − 2·15-s + 17-s − 2·19-s − 4·21-s − 8·23-s − 25-s + 27-s − 2·29-s + 4·31-s − 2·33-s + 8·35-s − 8·37-s + 39-s − 10·41-s − 2·45-s − 8·47-s + 9·49-s + 51-s + 2·53-s + 4·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.516·15-s + 0.242·17-s − 0.458·19-s − 0.872·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s + 1.35·35-s − 1.31·37-s + 0.160·39-s − 1.56·41-s − 0.298·45-s − 1.16·47-s + 9/7·49-s + 0.140·51-s + 0.274·53-s + 0.539·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.45508394571563, −14.76257482990762, −14.18512667299283, −13.46805956310570, −13.34012827839742, −12.65757622018912, −12.07519206419313, −11.86797249467420, −11.01665187281683, −10.33027607752053, −9.989438507798255, −9.575061647425097, −8.751622532594428, −8.325938537687093, −7.880547038329870, −7.236142016756150, −6.667525480893798, −6.172241996980991, −5.493922696011070, −4.698449970028522, −3.885714078183128, −3.623508454975790, −3.018609338239504, −2.316692509875232, −1.467161091398713, 0, 0,
1.467161091398713, 2.316692509875232, 3.018609338239504, 3.623508454975790, 3.885714078183128, 4.698449970028522, 5.493922696011070, 6.172241996980991, 6.667525480893798, 7.236142016756150, 7.880547038329870, 8.325938537687093, 8.751622532594428, 9.575061647425097, 9.989438507798255, 10.33027607752053, 11.01665187281683, 11.86797249467420, 12.07519206419313, 12.65757622018912, 13.34012827839742, 13.46805956310570, 14.18512667299283, 14.76257482990762, 15.45508394571563
