| L(s) = 1 | + 1.41i·2-s − 1.73i·3-s − 2.00·4-s + (−4.76 − 1.52i)5-s + 2.44·6-s + 5.00·7-s − 2.82i·8-s − 2.99·9-s + (2.16 − 6.73i)10-s + 0.0576i·11-s + 3.46i·12-s + 4.70i·13-s + 7.08i·14-s + (−2.64 + 8.24i)15-s + 4.00·16-s + 12.7·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (−0.952 − 0.305i)5-s + 0.408·6-s + 0.715·7-s − 0.353i·8-s − 0.333·9-s + (0.216 − 0.673i)10-s + 0.00524i·11-s + 0.288i·12-s + 0.362i·13-s + 0.506i·14-s + (−0.176 + 0.549i)15-s + 0.250·16-s + 0.749·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.283i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08299683308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08299683308\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73iT \) |
| 5 | \( 1 + (4.76 + 1.52i)T \) |
| 23 | \( 1 + (19.0 - 12.9i)T \) |
| good | 7 | \( 1 - 5.00T + 49T^{2} \) |
| 11 | \( 1 - 0.0576iT - 121T^{2} \) |
| 13 | \( 1 - 4.70iT - 169T^{2} \) |
| 17 | \( 1 - 12.7T + 289T^{2} \) |
| 19 | \( 1 + 27.4iT - 361T^{2} \) |
| 29 | \( 1 + 49.2T + 841T^{2} \) |
| 31 | \( 1 + 19.1T + 961T^{2} \) |
| 37 | \( 1 + 27.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 37.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 42.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 89.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 90.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 84.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 108. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 59.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 124.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 73.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 82.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 16.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 96.0T + 9.40e3T^{2} \) |
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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
−9.528354514988388264387805290129, −8.820717713565186195193314394679, −7.72233995039874062444396893423, −7.60221795509415705313731466881, −6.44668301665730247084517850136, −5.32682117721058739145465761019, −4.50365603213700301410956604805, −3.34460182906269229704190143221, −1.56370407289827706257184750943, −0.03013883232457812331923930445,
1.76757738084903982277070491663, 3.31440414541695847601053133000, 3.93404842750811220507341855532, 4.97463546740760489685833528007, 5.95867481011958794546011426796, 7.55181219713220031321394716995, 8.075061911354494087946212779336, 8.993552850812088261359361709688, 10.11821811858809123579377227491, 10.61042527472118103323318730880
