| L(s) = 1 | − 1.41i·2-s − 1.73·3-s − 2.00·4-s + (2.40 − 4.38i)5-s + 2.44i·6-s + (−6.94 − 0.853i)7-s + 2.82i·8-s + 2.99·9-s + (−6.20 − 3.39i)10-s + 2.88·11-s + 3.46·12-s − 13.8·13-s + (−1.20 + 9.82i)14-s + (−4.16 + 7.59i)15-s + 4.00·16-s − 24.1·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.500·4-s + (0.480 − 0.876i)5-s + 0.408i·6-s + (−0.992 − 0.121i)7-s + 0.353i·8-s + 0.333·9-s + (−0.620 − 0.339i)10-s + 0.261·11-s + 0.288·12-s − 1.06·13-s + (−0.0861 + 0.701i)14-s + (−0.277 + 0.506i)15-s + 0.250·16-s − 1.42·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.104933 + 0.325548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.104933 + 0.325548i\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73T \) |
| 5 | \( 1 + (-2.40 + 4.38i)T \) |
| 7 | \( 1 + (6.94 + 0.853i)T \) |
| good | 11 | \( 1 - 2.88T + 121T^{2} \) |
| 13 | \( 1 + 13.8T + 169T^{2} \) |
| 17 | \( 1 + 24.1T + 289T^{2} \) |
| 19 | \( 1 - 6.53iT - 361T^{2} \) |
| 23 | \( 1 - 28.8iT - 529T^{2} \) |
| 29 | \( 1 + 32.9T + 841T^{2} \) |
| 31 | \( 1 + 2.43iT - 961T^{2} \) |
| 37 | \( 1 + 50.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 21.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 17.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 120. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 21.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 66.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 78.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 90.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 44.1T + 9.40e3T^{2} \) |
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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
−11.66648240570342748313940280368, −10.59419846896465444328662880608, −9.535704544573008569607603532698, −9.120528838341404368541101902543, −7.45148346392138254749170559664, −6.13305341357252059476220975708, −5.07226749934198671552634503448, −3.86993999111919272512399108850, −2.04299729567288167162465157691, −0.19324878262577795728968904115,
2.64250081782926498525227138006, 4.35262718232220988986193196154, 5.73649248418936732517703170169, 6.65610123855452336071119809489, 7.18472582659355016955026973632, 8.874377099953731328529851858814, 9.806300804829929493660775464723, 10.60663711318368774584855069859, 11.77902426061976557983403131325, 12.90636621463881902651451125067
