L(s) = 1 | − 2.46i·2-s + 1.92·4-s + (0.0773 + 11.1i)5-s + 19.2i·7-s − 24.4i·8-s + (27.5 − 0.190i)10-s − 39.8·11-s + 1.00i·13-s + 47.4·14-s − 44.8·16-s + 52.6i·17-s + 49.5·19-s + (0.149 + 21.5i)20-s + 98.1i·22-s − 27.4i·23-s + ⋯ |
L(s) = 1 | − 0.871i·2-s + 0.241·4-s + (0.00692 + 0.999i)5-s + 1.03i·7-s − 1.08i·8-s + (0.871 − 0.00602i)10-s − 1.09·11-s + 0.0215i·13-s + 0.904·14-s − 0.700·16-s + 0.750i·17-s + 0.598·19-s + (0.00166 + 0.241i)20-s + 0.951i·22-s − 0.248i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00692 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00692 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.120046710\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120046710\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.0773 - 11.1i)T \) |
good | 2 | \( 1 + 2.46iT - 8T^{2} \) |
| 7 | \( 1 - 19.2iT - 343T^{2} \) |
| 11 | \( 1 + 39.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.00iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 52.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 49.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 419. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 398.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 358. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 141. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 290. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 28.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 732.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 176. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 802.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 512. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 612.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 80.8iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 24.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.36e3iT - 9.12e5T^{2} \) |
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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
−11.20497905152220165145443868951, −10.25850499928802653349582641842, −9.656824458719266580114302267913, −8.330828994240832443311855665776, −7.30920412943717387861663607333, −6.31457441074522453339388339175, −5.33050730681123638358869130159, −3.60281979623459889005162118802, −2.74747834069367055212028796321, −1.83968987236599317078043388112,
0.33554698057630893631387919997, 2.01156321733132860449882549978, 3.73449254896809411126574540355, 5.13858021999553088706336022545, 5.60787101481581353283902503551, 7.20714950897389730891066214066, 7.50486204377054708593178036512, 8.546874418180277882776910790712, 9.549087197382180959775603748149, 10.64710053131372178025657955211