| 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
−12.69267043965630418369567781394, −11.51145452764345310536263030176, −10.54031178378514465236774573840, −9.483267814542858491651289402308, −8.369491442889014870418804495325, −6.82982408786385634283016123182, −6.13713334750910270506618633621, −5.03090508308574517731895224206, −3.52902477250223485020805119097, −0.30699216331944433983331239245,
1.85558413461084730612344733981, 3.88384841633145044921650500870, 5.18108163942993541853823265893, 6.48397424181943089329067829077, 7.56214326176488323051394462331, 9.278759661049249007746470351485, 10.14677161441246214606648929390, 10.95987585118383088875825118490, 12.20719353362292631505051888905, 12.62228449396781183689940472090
