L(s) = 1 | − 1.41i·2-s + (−2.91 − 0.707i)3-s − 2.00·4-s + (−1.00 + 4.12i)6-s − 5.83·7-s + 2.82i·8-s + (8 + 4.12i)9-s + 16.4i·11-s + (5.83 + 1.41i)12-s + 8.24i·14-s + 4.00·16-s + 11.3i·17-s + (5.83 − 11.3i)18-s − 12·19-s + (17 + 4.12i)21-s + 23.3·22-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.971 − 0.235i)3-s − 0.500·4-s + (−0.166 + 0.687i)6-s − 0.832·7-s + 0.353i·8-s + (0.888 + 0.458i)9-s + 1.49i·11-s + (0.485 + 0.117i)12-s + 0.589i·14-s + 0.250·16-s + 0.665i·17-s + (0.323 − 0.628i)18-s − 0.631·19-s + (0.809 + 0.196i)21-s + 1.06·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.337381 + 0.265335i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.337381 + 0.265335i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (2.91 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 5.83T + 49T^{2} \) |
| 11 | \( 1 - 16.4iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 11.3iT - 289T^{2} \) |
| 19 | \( 1 + 12T + 361T^{2} \) |
| 23 | \( 1 - 24.0iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 32T + 961T^{2} \) |
| 37 | \( 1 + 23.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 57.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 67.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16T + 3.72e3T^{2} \) |
| 67 | \( 1 - 5.83T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 116.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72T + 6.24e3T^{2} \) |
| 83 | \( 1 + 43.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 65.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 163.T + 9.40e3T^{2} \) |
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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
−12.62228449396781183689940472090, −12.20719353362292631505051888905, −10.95987585118383088875825118490, −10.14677161441246214606648929390, −9.278759661049249007746470351485, −7.56214326176488323051394462331, −6.48397424181943089329067829077, −5.18108163942993541853823265893, −3.88384841633145044921650500870, −1.85558413461084730612344733981,
0.30699216331944433983331239245, 3.52902477250223485020805119097, 5.03090508308574517731895224206, 6.13713334750910270506618633621, 6.82982408786385634283016123182, 8.369491442889014870418804495325, 9.483267814542858491651289402308, 10.54031178378514465236774573840, 11.51145452764345310536263030176, 12.69267043965630418369567781394