L(s) = 1 | + (−2.73 + 0.732i)2-s + (−9.68 + 5.59i)3-s + (6.92 − 4i)4-s + (−24.8 − 6.64i)5-s + (22.3 − 22.3i)6-s + (−19.9 − 34.4i)7-s + (−15.9 + 16i)8-s + (22.0 − 38.1i)9-s + 72.6·10-s + 64.8i·11-s + (−44.7 + 77.4i)12-s + (111. + 29.9i)13-s + (79.6 + 79.6i)14-s + (277. − 74.3i)15-s + (31.9 − 55.4i)16-s + (−15.2 − 57.0i)17-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (−1.07 + 0.621i)3-s + (0.433 − 0.250i)4-s + (−0.992 − 0.265i)5-s + (0.621 − 0.621i)6-s + (−0.406 − 0.703i)7-s + (−0.249 + 0.250i)8-s + (0.271 − 0.470i)9-s + 0.726·10-s + 0.536i·11-s + (−0.310 + 0.537i)12-s + (0.661 + 0.177i)13-s + (0.406 + 0.406i)14-s + (1.23 − 0.330i)15-s + (0.124 − 0.216i)16-s + (−0.0529 − 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0769i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 - 0.0769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.547912 + 0.0211177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.547912 + 0.0211177i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.73 - 0.732i)T \) |
| 37 | \( 1 + (17.1 + 1.36e3i)T \) |
good | 3 | \( 1 + (9.68 - 5.59i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (24.8 + 6.64i)T + (541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (19.9 + 34.4i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 64.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-111. - 29.9i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (15.2 + 57.0i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-628. - 168. i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-152. + 152. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-312. - 312. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (68.3 + 68.3i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (1.70e3 - 984. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-908. + 908. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 2.23e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (22.0 - 38.2i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.07e3 + 4.01e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (307. - 1.14e3i)T + (-1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-6.82e3 + 3.93e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-4.03e3 - 6.99e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 1.31e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-9.16e3 - 2.45e3i)T + (3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-1.30e3 + 2.26e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-4.27e3 + 1.14e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (5.67e3 - 5.67e3i)T - 8.85e7iT^{2} \) |
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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
−13.94067335166264238114765283982, −12.29163446962157608287795616502, −11.43214834523673715006564233794, −10.53164737173966025548388354602, −9.505633345579998811267199036504, −7.964876545149829091586132282791, −6.82397079346257415786125445069, −5.28628943380072563076982000666, −3.86036041347009890465213802059, −0.64210389279915573976668283857,
0.834632805836137244273669197466, 3.26785504150180704454644135416, 5.60007335857681757360032348193, 6.76846351160832924677260205085, 7.912388500379089066506086351727, 9.224502722313432339600411940639, 10.81251456972377573607336010654, 11.66041073829049732564307211154, 12.17800671696888025915577817769, 13.52205163811933571229122587872