L(s) = 1 | − i·2-s + 3-s − 4-s − i·5-s − i·6-s − 4.80i·7-s + i·8-s + 9-s − 10-s − 2.55i·11-s − 12-s − 4.80·14-s − i·15-s + 16-s + 3.02·17-s − i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 1.81i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.770i·11-s − 0.288·12-s − 1.28·14-s − 0.258i·15-s + 0.250·16-s + 0.734·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.234388232\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.234388232\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 4.80iT - 7T^{2} \) |
| 11 | \( 1 + 2.55iT - 11T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 + 3.89iT - 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 - 5.44iT - 31T^{2} \) |
| 37 | \( 1 + 7.51iT - 37T^{2} \) |
| 41 | \( 1 + 5.89iT - 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 6.42iT - 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 + 7.60iT - 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 0.929iT - 67T^{2} \) |
| 71 | \( 1 - 14.9iT - 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 5.40T + 79T^{2} \) |
| 83 | \( 1 - 4.33iT - 83T^{2} \) |
| 89 | \( 1 - 16.8iT - 89T^{2} \) |
| 97 | \( 1 - 17.9iT - 97T^{2} \) |
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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
−7.971982403271934515787959402319, −7.27617785855618452015490769567, −6.63709283889020826384918550694, −5.43629867612921401370716544008, −4.69873405430538934203655818283, −3.88680264408018960750061816419, −3.44032922867463791936741232613, −2.44782463179688390390343418303, −1.17181104551611772204828101335, −0.62061807626077117394380634180,
1.52404693768390337656619097040, 2.56835891833709634066077370664, 3.12969651618580874200166212718, 4.27688525027894986172771287928, 5.03192701170317853579303953829, 5.91887619298224809967256700326, 6.27443176037765733499698891669, 7.28126795518163592444351890915, 7.930868801918087454452386864180, 8.448320642899004409904954787448