L(s) = 1 | − 2.29·2-s − 3-s + 3.26·4-s − 3.12·5-s + 2.29·6-s − 2.80·7-s − 2.89·8-s + 9-s + 7.17·10-s + 0.416·11-s − 3.26·12-s − 13-s + 6.42·14-s + 3.12·15-s + 0.121·16-s + 6.72·17-s − 2.29·18-s − 0.991·19-s − 10.2·20-s + 2.80·21-s − 0.955·22-s + 5.08·23-s + 2.89·24-s + 4.77·25-s + 2.29·26-s − 27-s − 9.14·28-s + ⋯ |
L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.63·4-s − 1.39·5-s + 0.936·6-s − 1.05·7-s − 1.02·8-s + 0.333·9-s + 2.26·10-s + 0.125·11-s − 0.941·12-s − 0.277·13-s + 1.71·14-s + 0.807·15-s + 0.0303·16-s + 1.63·17-s − 0.540·18-s − 0.227·19-s − 2.28·20-s + 0.611·21-s − 0.203·22-s + 1.06·23-s + 0.591·24-s + 0.955·25-s + 0.449·26-s − 0.192·27-s − 1.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2320785179\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2320785179\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 5 | \( 1 + 3.12T + 5T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 - 0.416T + 11T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 + 0.991T + 19T^{2} \) |
| 23 | \( 1 - 5.08T + 23T^{2} \) |
| 29 | \( 1 + 2.63T + 29T^{2} \) |
| 31 | \( 1 - 1.49T + 31T^{2} \) |
| 37 | \( 1 + 2.62T + 37T^{2} \) |
| 41 | \( 1 - 0.0147T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 + 5.68T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 + 6.67T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 + 1.69T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 6.49T + 97T^{2} \) |
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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
−8.373322561536142475804157080383, −7.80446349922077992519883878773, −7.13948114251751926104187738084, −6.70367662449184424608141524992, −5.70634421403031483068662516580, −4.66755709320186568678275629114, −3.61189405851063058337633313092, −2.92740292490835551309350605447, −1.38970257477398769428939104613, −0.39048458249261433358635477577,
0.39048458249261433358635477577, 1.38970257477398769428939104613, 2.92740292490835551309350605447, 3.61189405851063058337633313092, 4.66755709320186568678275629114, 5.70634421403031483068662516580, 6.70367662449184424608141524992, 7.13948114251751926104187738084, 7.80446349922077992519883878773, 8.373322561536142475804157080383