| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.64i·7-s + 2.82·8-s + 20.8i·11-s − 5.64i·13-s − 3.74i·14-s + 4.00·16-s − 16.5·17-s + 1.77·19-s + 29.5i·22-s + 23.6·23-s − 7.98i·26-s − 5.29i·28-s + 29.3i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.377i·7-s + 0.353·8-s + 1.89i·11-s − 0.434i·13-s − 0.267i·14-s + 0.250·16-s − 0.974·17-s + 0.0932·19-s + 1.34i·22-s + 1.02·23-s − 0.307i·26-s − 0.188i·28-s + 1.01i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.028458582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028458582\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 - 20.8iT - 121T^{2} \) |
| 13 | \( 1 + 5.64iT - 169T^{2} \) |
| 17 | \( 1 + 16.5T + 289T^{2} \) |
| 19 | \( 1 - 1.77T + 361T^{2} \) |
| 23 | \( 1 - 23.6T + 529T^{2} \) |
| 29 | \( 1 - 29.3iT - 841T^{2} \) |
| 31 | \( 1 + 17.1T + 961T^{2} \) |
| 37 | \( 1 + 14.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 29.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 39.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 19.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 88.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 85.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 19.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 58.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 113. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 14.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 151.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 15.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 57.2iT - 9.40e3T^{2} \) |
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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
−8.810985082089372224026363950092, −7.71567769718767857999155881687, −7.10633301664593556367365243821, −6.66653275536000470436724880658, −5.53902318834859380345923512266, −4.77795999674730300644250716197, −4.27224870992164556122777804703, −3.23962389531059273158317880146, −2.28437438408403146471104870312, −1.36134029148652114963581534628,
0.32286681022108568926042910688, 1.66176364921481284268767940848, 2.80169348437225585498413309762, 3.41509110418681137527932812684, 4.40524434717681982556520062394, 5.18991045056527847340090414479, 6.05435153612230216046863751738, 6.45307516812499317744869247708, 7.43056046879660805952267578850, 8.343185488762850861038409968215
