L(s) = 1 | − 3·3-s − 4·7-s + 6·9-s − 2·11-s + 5·13-s − 4·17-s + 2·19-s + 12·21-s + 23-s − 9·27-s − 7·29-s + 3·31-s + 6·33-s − 2·37-s − 15·39-s − 9·41-s − 8·43-s + 9·47-s + 9·49-s + 12·51-s − 2·53-s − 6·57-s − 2·61-s − 24·63-s + 14·67-s − 3·69-s + 3·71-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1.51·7-s + 2·9-s − 0.603·11-s + 1.38·13-s − 0.970·17-s + 0.458·19-s + 2.61·21-s + 0.208·23-s − 1.73·27-s − 1.29·29-s + 0.538·31-s + 1.04·33-s − 0.328·37-s − 2.40·39-s − 1.40·41-s − 1.21·43-s + 1.31·47-s + 9/7·49-s + 1.68·51-s − 0.274·53-s − 0.794·57-s − 0.256·61-s − 3.02·63-s + 1.71·67-s − 0.361·69-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−6.90035333980951981978937222896, −6.66594669567581980430729862767, −6.02771573886787652498895446151, −5.47556142437483889479834900997, −4.81024105608428314617771606455, −3.83476933689227639877998666629, −3.29336370154811067963272498287, −2.01206047983172759500763124469, −0.832565523311458848680696537063, 0,
0.832565523311458848680696537063, 2.01206047983172759500763124469, 3.29336370154811067963272498287, 3.83476933689227639877998666629, 4.81024105608428314617771606455, 5.47556142437483889479834900997, 6.02771573886787652498895446151, 6.66594669567581980430729862767, 6.90035333980951981978937222896