| L(s) = 1 | + 5.47i·2-s − 46.6·3-s + 98.0·4-s + 442.·5-s − 255. i·6-s − 480. i·7-s + 1.23e3i·8-s − 10.3·9-s + 2.41e3i·10-s − 7.82e3i·11-s − 4.57e3·12-s − 4.44e3·13-s + 2.63e3·14-s − 2.06e4·15-s + 5.78e3·16-s − 597. i·17-s + ⋯ |
| L(s) = 1 | + 0.483i·2-s − 0.997·3-s + 0.766·4-s + 1.58·5-s − 0.482i·6-s − 0.529i·7-s + 0.853i·8-s − 0.00471·9-s + 0.764i·10-s − 1.77i·11-s − 0.764·12-s − 0.560·13-s + 0.256·14-s − 1.57·15-s + 0.353·16-s − 0.0294i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.03769 - 0.277461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03769 - 0.277461i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (-1.70e6 - 4.73e5i)T \) |
| good | 2 | \( 1 - 5.47iT - 128T^{2} \) |
| 3 | \( 1 + 46.6T + 2.18e3T^{2} \) |
| 5 | \( 1 - 442.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 480. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 7.82e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 4.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 597. iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 3.36e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.51e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 9.87e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 7.30e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 5.80e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.88e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.67e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 5.62e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.84e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.24e6iT - 2.48e12T^{2} \) |
| 67 | \( 1 + 9.81e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 3.73e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 4.96e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.29e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 1.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.01e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 8.27e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.81381574938188326841669894673, −12.29990563560366220324101677888, −11.02156718067137816244896182096, −10.42985827710038174718811109596, −8.795444235568935330940074042667, −7.03603631064957109086301091167, −5.93873868076399708897959506227, −5.43327797980084015811502189336, −2.69259490913439056969150391085, −0.894038664005839591192187624225,
1.48156992853543001874546923709, 2.55234731083437462245640746952, 5.13172551422472997589508968363, 6.06923253680122205807559605727, 7.21461675446625061066907562793, 9.654759761119687559061988662736, 10.05296364684616959239865052608, 11.52338246151120362036170868871, 12.20640871498137577867369797678, 13.31629661972320787694878785485
