L(s) = 1 | + 2·2-s + (1.53 − 4.96i)3-s + 4·4-s − 1.00i·5-s + (3.06 − 9.93i)6-s + 8.17·7-s + 8·8-s + (−22.3 − 15.1i)9-s − 2.01i·10-s − 49.9·11-s + (6.12 − 19.8i)12-s − 46.3i·13-s + 16.3·14-s + (−5.01 − 1.54i)15-s + 16·16-s − 111. i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.294 − 0.955i)3-s + 0.5·4-s − 0.0903i·5-s + (0.208 − 0.675i)6-s + 0.441·7-s + 0.353·8-s + (−0.826 − 0.562i)9-s − 0.0638i·10-s − 1.36·11-s + (0.147 − 0.477i)12-s − 0.988i·13-s + 0.311·14-s + (−0.0863 − 0.0265i)15-s + 0.250·16-s − 1.58i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.660106148\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.660106148\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + (-1.53 + 4.96i)T \) |
| 59 | \( 1 + (-431. - 138. i)T \) |
good | 5 | \( 1 + 1.00iT - 125T^{2} \) |
| 7 | \( 1 - 8.17T + 343T^{2} \) |
| 11 | \( 1 + 49.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 111. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 48.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 14.5iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 2.36iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 127. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 82.9iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 39.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 434. iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 572. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 265. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 491. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 841. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 816.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 854. iT - 9.12e5T^{2} \) |
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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
−10.99587027691725013877721825510, −9.912616713928132722189138725772, −8.524195483027692484189359443793, −7.70617302927485193243347813351, −7.01856486106983798811812144569, −5.64062069050951759017701563032, −4.98873935045754259394563459924, −3.19918347895484989832382648191, −2.34991427864055608625308429010, −0.67600045852327042076501442291,
2.05781807361017495531358244703, 3.30415800622204361196361859403, 4.40523209774273885695654724359, 5.21379333441203594049305547785, 6.27720031677221234218340297271, 7.72696572596304390769229981820, 8.481782831252408321245134999907, 9.700204803973517062428127912558, 10.62549113361908859130914170627, 11.16095586497358169016552264569