| L(s) = 1 | + (1.41 − 2.44i)2-s + (−3.99 − 6.92i)4-s + (−21.9 − 12.6i)5-s + (−7 + 48.4i)7-s − 22.6·8-s + (−62.1 + 35.9i)10-s + (21.9 + 38.0i)11-s + 162. i·13-s + (108. + 85.7i)14-s + (−32.0 + 55.4i)16-s + (−93.7 + 54.1i)17-s + (389. + 224. i)19-s + 203. i·20-s + 124.·22-s + (−217. + 377. i)23-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.879 − 0.507i)5-s + (−0.142 + 0.989i)7-s − 0.353·8-s + (−0.621 + 0.359i)10-s + (0.181 + 0.314i)11-s + 0.961i·13-s + (0.555 + 0.437i)14-s + (−0.125 + 0.216i)16-s + (−0.324 + 0.187i)17-s + (1.07 + 0.622i)19-s + 0.507i·20-s + 0.256·22-s + (−0.411 + 0.713i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.733135 + 0.557736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.733135 + 0.557736i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 2.44i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (7 - 48.4i)T \) |
| good | 5 | \( 1 + (21.9 + 12.6i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (-21.9 - 38.0i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 162. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (93.7 - 54.1i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-389. - 224. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (217. - 377. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + 742.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (900. - 519. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (493. - 854. i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 1.14e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.41e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (-1.16e3 - 672. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (1.78e3 + 3.08e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (3.71e3 - 2.14e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.20e3 + 698. i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.63e3 + 6.29e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 5.98e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (500. - 288. i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.57e3 + 2.72e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 4.72e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-725. - 418. i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 5.62e3iT - 8.85e7T^{2} \) |
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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
−12.52233961533074369907794556169, −11.97412067553098577226125213916, −11.19869409529417231238185104765, −9.646941756340093542635970844631, −8.866249361561167734611652187958, −7.56965561997432072848876709785, −5.96161419763693693373347792133, −4.68676026331060835200944169048, −3.48126444802993689901658523548, −1.74031593869251843742930281111,
0.34575469021196868838455809089, 3.21058222920407238829783308329, 4.23510753861796667746183380517, 5.76675728676495871922716618321, 7.23627544153522515157731880722, 7.65370036214452377938444927827, 9.120028910386365072368919457758, 10.56641019346932968950549307431, 11.39863639129638616295135542039, 12.62928920669691867551238257628
