| L(s) = 1 | − 0.835·2-s − 3.30·4-s + (1.17 + 4.86i)5-s − 10.8i·7-s + 6.10·8-s + (−0.981 − 4.06i)10-s − 3.31i·11-s + 21.7i·13-s + 9.04i·14-s + 8.10·16-s + 1.02·17-s − 24.6·19-s + (−3.87 − 16.0i)20-s + 2.77i·22-s − 1.45·23-s + ⋯ |
| L(s) = 1 | − 0.417·2-s − 0.825·4-s + (0.234 + 0.972i)5-s − 1.54i·7-s + 0.762·8-s + (−0.0981 − 0.406i)10-s − 0.301i·11-s + 1.67i·13-s + 0.646i·14-s + 0.506·16-s + 0.0600·17-s − 1.29·19-s + (−0.193 − 0.802i)20-s + 0.126i·22-s − 0.0633·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5672796471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5672796471\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.17 - 4.86i)T \) |
| 11 | \( 1 + 3.31iT \) |
| good | 2 | \( 1 + 0.835T + 4T^{2} \) |
| 7 | \( 1 + 10.8iT - 49T^{2} \) |
| 13 | \( 1 - 21.7iT - 169T^{2} \) |
| 17 | \( 1 - 1.02T + 289T^{2} \) |
| 19 | \( 1 + 24.6T + 361T^{2} \) |
| 23 | \( 1 + 1.45T + 529T^{2} \) |
| 29 | \( 1 + 56.3iT - 841T^{2} \) |
| 31 | \( 1 - 1.48T + 961T^{2} \) |
| 37 | \( 1 + 48.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 51.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 74.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 84.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 42.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 11.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 103.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 54.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 49.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 62.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 52.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 31.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 50.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 110. iT - 9.40e3T^{2} \) |
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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
−10.54398313201719845813514606064, −9.619444206583794071088341278932, −8.811319337675920554890709817911, −7.61857018077063918547803615182, −7.00062929725273671460328014373, −5.96361575989690543927529424855, −4.25095547657853240565528493354, −3.93302390174369051626233134475, −2.00722717875744365861977088957, −0.28548606734916549707698231479,
1.37251759092701673137043099246, 2.93798767830890701420588267311, 4.59366516268919144629849887067, 5.28804649206647071543908640160, 6.12823548880621553465328761574, 7.892254332867404309648143595063, 8.481128895510163055934910951261, 9.091902122194807972705934299490, 9.891176298206372262579198482475, 10.79680912608564771437281776781
