| L(s) = 1 | + (−1.29 + 2.70i)3-s + (−1.73 + 3.00i)7-s + (−5.66 − 6.99i)9-s + (−12.3 − 7.14i)11-s + (−3.04 − 5.28i)13-s + 11.7i·17-s + 23.4·19-s + (−5.88 − 8.56i)21-s + (24.8 − 14.3i)23-s + (26.2 − 6.30i)27-s + (−13.3 − 7.68i)29-s + (−18.4 − 32.0i)31-s + (35.3 − 24.2i)33-s + 46.7·37-s + (18.2 − 1.43i)39-s + ⋯ |
| L(s) = 1 | + (−0.430 + 0.902i)3-s + (−0.247 + 0.428i)7-s + (−0.629 − 0.777i)9-s + (−1.12 − 0.649i)11-s + (−0.234 − 0.406i)13-s + 0.690i·17-s + 1.23·19-s + (−0.280 − 0.407i)21-s + (1.08 − 0.624i)23-s + (0.972 − 0.233i)27-s + (−0.459 − 0.265i)29-s + (−0.596 − 1.03i)31-s + (1.07 − 0.735i)33-s + 1.26·37-s + (0.467 − 0.0368i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.183398921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183398921\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.29 - 2.70i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (1.73 - 3.00i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (12.3 + 7.14i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (3.04 + 5.28i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 11.7iT - 289T^{2} \) |
| 19 | \( 1 - 23.4T + 361T^{2} \) |
| 23 | \( 1 + (-24.8 + 14.3i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (13.3 + 7.68i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (18.4 + 32.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 46.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (17.1 - 9.89i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-2.75 + 4.76i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (8.93 + 5.15i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 41.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-58.5 + 33.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (26.5 - 45.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (15.9 + 27.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 2.28iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 86.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-58.4 + 101. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-113. - 65.2i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 75.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-14.1 + 24.4i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.906222253040580673938289896538, −9.252384709910718864666379127053, −8.335944634230286365594407217718, −7.45437303906140921214222316571, −6.10604560210536691845608552773, −5.54875498921837744566913299743, −4.70583010824221063158768610877, −3.47424170103291418282377730894, −2.64550808404384215996091531137, −0.55046309178106700521461123805,
0.882686066088126004938369400264, 2.22565809083597926451817845502, 3.32137939370321009625593245604, 4.95065101919584820335247408959, 5.40461205753906296177334561869, 6.75538891045746343645112461086, 7.29085736916916511589727518598, 7.908577933770330291602779284108, 9.131185057065842186694619366912, 9.944649276444461294888555416911
