| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 4·11-s + 13-s + 15-s − 17-s − 2·19-s + 2·21-s + 25-s − 27-s + 2·29-s − 4·33-s + 2·35-s − 2·37-s − 39-s + 2·41-s + 6·43-s − 45-s − 12·47-s − 3·49-s + 51-s + 10·53-s − 4·55-s + 2·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.458·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.160·39-s + 0.312·41-s + 0.914·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s + 0.140·51-s + 1.37·53-s − 0.539·55-s + 0.264·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.56857210989540, −15.13581128217974, −14.52114159537778, −13.97954267710378, −13.32566100596376, −12.80184235548013, −12.29839623910536, −11.81748899718445, −11.28937008261187, −10.78981660485607, −10.16071314887553, −9.545816373709085, −9.077661745618550, −8.444294067870946, −7.823916699726954, −6.943023009025594, −6.679312703982764, −6.122832355135305, −5.473838853688823, −4.627771477346264, −4.081259870259880, −3.536537693277226, −2.766185836987251, −1.755284096478582, −0.9303324138170849, 0,
0.9303324138170849, 1.755284096478582, 2.766185836987251, 3.536537693277226, 4.081259870259880, 4.627771477346264, 5.473838853688823, 6.122832355135305, 6.679312703982764, 6.943023009025594, 7.823916699726954, 8.444294067870946, 9.077661745618550, 9.545816373709085, 10.16071314887553, 10.78981660485607, 11.28937008261187, 11.81748899718445, 12.29839623910536, 12.80184235548013, 13.32566100596376, 13.97954267710378, 14.52114159537778, 15.13581128217974, 15.56857210989540
