| L(s) = 1 | − 3-s − 5·7-s + 9-s − 11-s − 13-s + 3·17-s − 6·19-s + 5·21-s − 7·23-s − 27-s + 6·29-s + 2·31-s + 33-s − 37-s + 39-s + 7·41-s + 8·43-s + 2·47-s + 18·49-s − 3·51-s − 13·53-s + 6·57-s + 8·59-s − 7·61-s − 5·63-s + 12·67-s + 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.727·17-s − 1.37·19-s + 1.09·21-s − 1.45·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.174·33-s − 0.164·37-s + 0.160·39-s + 1.09·41-s + 1.21·43-s + 0.291·47-s + 18/7·49-s − 0.420·51-s − 1.78·53-s + 0.794·57-s + 1.04·59-s − 0.896·61-s − 0.629·63-s + 1.46·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−16.16528739487570, −15.84573049463830, −15.48455355498474, −14.53832393301949, −14.05759225578918, −13.37220410127848, −12.76706187919164, −12.37493035969739, −12.10763196487477, −11.12584792907028, −10.48803774335955, −10.10996010630279, −9.584934995065891, −9.025030185706855, −8.131711847595937, −7.601738226226569, −6.724800262407068, −6.289066098914923, −5.944035225123106, −5.090372026803209, −4.209054435894281, −3.700286120278077, −2.811903727930929, −2.197333670358173, −0.8165336003507947, 0,
0.8165336003507947, 2.197333670358173, 2.811903727930929, 3.700286120278077, 4.209054435894281, 5.090372026803209, 5.944035225123106, 6.289066098914923, 6.724800262407068, 7.601738226226569, 8.131711847595937, 9.025030185706855, 9.584934995065891, 10.10996010630279, 10.48803774335955, 11.12584792907028, 12.10763196487477, 12.37493035969739, 12.76706187919164, 13.37220410127848, 14.05759225578918, 14.53832393301949, 15.48455355498474, 15.84573049463830, 16.16528739487570
