L(s) = 1 | + 3-s − 2·4-s − 1.73·7-s + 9-s − 2·12-s + 4·16-s + 3.46·17-s − 5.19·19-s − 1.73·21-s − 6·23-s + 27-s + 3.46·28-s + 6.92·29-s + 31-s − 2·36-s + 5·37-s + 3.46·41-s − 10.3·43-s + 12·47-s + 4·48-s − 4·49-s + 3.46·51-s − 6·53-s − 5.19·57-s − 12.1·61-s − 1.73·63-s − 8·64-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.654·7-s + 0.333·9-s − 0.577·12-s + 16-s + 0.840·17-s − 1.19·19-s − 0.377·21-s − 1.25·23-s + 0.192·27-s + 0.654·28-s + 1.28·29-s + 0.179·31-s − 0.333·36-s + 0.821·37-s + 0.541·41-s − 1.58·43-s + 1.75·47-s + 0.577·48-s − 0.571·49-s + 0.485·51-s − 0.824·53-s − 0.688·57-s − 1.55·61-s − 0.218·63-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.436641876\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.436641876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−8.032054782214698978927112473398, −7.13711717080999035317305935993, −6.22958245868092253509088806722, −5.80486623397408439772546422653, −4.69339965261509922777148182097, −4.27044203224783758105407073860, −3.45397851510919041401784544714, −2.81981699177738900059336227282, −1.72958462479312913654454533508, −0.56602574337960999418448541241,
0.56602574337960999418448541241, 1.72958462479312913654454533508, 2.81981699177738900059336227282, 3.45397851510919041401784544714, 4.27044203224783758105407073860, 4.69339965261509922777148182097, 5.80486623397408439772546422653, 6.22958245868092253509088806722, 7.13711717080999035317305935993, 8.032054782214698978927112473398