L(s) = 1 | − 1.32·2-s − 3-s − 0.246·4-s − 4.03·5-s + 1.32·6-s + 2.97·8-s + 9-s + 5.34·10-s + 4.32·11-s + 0.246·12-s − 1.67·13-s + 4.03·15-s − 3.44·16-s + 1.53·17-s − 1.32·18-s − 4.63·19-s + 0.995·20-s − 5.72·22-s + 1.68·23-s − 2.97·24-s + 11.2·25-s + 2.22·26-s − 27-s + 6.19·29-s − 5.34·30-s + 7.82·31-s − 1.38·32-s + ⋯ |
L(s) = 1 | − 0.936·2-s − 0.577·3-s − 0.123·4-s − 1.80·5-s + 0.540·6-s + 1.05·8-s + 0.333·9-s + 1.68·10-s + 1.30·11-s + 0.0712·12-s − 0.465·13-s + 1.04·15-s − 0.861·16-s + 0.372·17-s − 0.312·18-s − 1.06·19-s + 0.222·20-s − 1.21·22-s + 0.350·23-s − 0.607·24-s + 2.25·25-s + 0.435·26-s − 0.192·27-s + 1.14·29-s − 0.975·30-s + 1.40·31-s − 0.245·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5350200432\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5350200432\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 5 | \( 1 + 4.03T + 5T^{2} \) |
| 11 | \( 1 - 4.32T + 11T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 + 4.63T + 19T^{2} \) |
| 23 | \( 1 - 1.68T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 - 7.25T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 6.75T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 5.99T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 - 0.0883T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 2.82T + 97T^{2} \) |
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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
−8.099779197236862024013229311631, −7.53789142180245225770150504849, −6.86719726196017621867633600808, −6.24380099413500542959669291409, −4.93591234270137903775168290750, −4.29722735195252997189487270019, −3.99533828239259021437307215552, −2.78103730441282172909348071082, −1.23704346066914761973163286684, −0.54489654852909615213263963486,
0.54489654852909615213263963486, 1.23704346066914761973163286684, 2.78103730441282172909348071082, 3.99533828239259021437307215552, 4.29722735195252997189487270019, 4.93591234270137903775168290750, 6.24380099413500542959669291409, 6.86719726196017621867633600808, 7.53789142180245225770150504849, 8.099779197236862024013229311631