| L(s) = 1 | + 1.41·2-s − 1.48·3-s − 1.98·4-s − 2.10·6-s + 2.50i·7-s − 8.49·8-s − 6.80·9-s + 16.7·11-s + 2.94·12-s + 12.8·13-s + 3.55i·14-s − 4.09·16-s − 10.1i·17-s − 9.65·18-s + (−2.53 − 18.8i)19-s + ⋯ |
| L(s) = 1 | + 0.709·2-s − 0.493·3-s − 0.497·4-s − 0.350·6-s + 0.357i·7-s − 1.06·8-s − 0.756·9-s + 1.52·11-s + 0.245·12-s + 0.991·13-s + 0.253i·14-s − 0.255·16-s − 0.595i·17-s − 0.536·18-s + (−0.133 − 0.991i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.525155625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525155625\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (2.53 + 18.8i)T \) |
| good | 2 | \( 1 - 1.41T + 4T^{2} \) |
| 3 | \( 1 + 1.48T + 9T^{2} \) |
| 7 | \( 1 - 2.50iT - 49T^{2} \) |
| 11 | \( 1 - 16.7T + 121T^{2} \) |
| 13 | \( 1 - 12.8T + 169T^{2} \) |
| 17 | \( 1 + 10.1iT - 289T^{2} \) |
| 23 | \( 1 + 18.7iT - 529T^{2} \) |
| 29 | \( 1 + 30.4iT - 841T^{2} \) |
| 31 | \( 1 + 56.8iT - 961T^{2} \) |
| 37 | \( 1 - 19.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 15.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 68.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 33.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.77T + 2.80e3T^{2} \) |
| 59 | \( 1 + 41.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 80.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 73.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 119. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 66.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 56.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 124. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 9.31iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 7.61T + 9.40e3T^{2} \) |
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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
−11.10544316613697288120229144119, −9.488988935725528913649884125340, −9.054706308350196446667596400891, −8.059650915397740440045555514615, −6.30078353567758670882235086456, −6.13419277699630129129423476227, −4.82813114125948219742936693396, −4.00203334070084475996571352785, −2.75213859948002234992897916589, −0.63345786407053604720839448621,
1.24203545733339542301798044741, 3.44011415642611444967638782902, 4.01374088664094888615250680630, 5.34313395972922566951213195404, 6.03037006639065233016298964757, 6.92253382151577563648562570985, 8.557613961690949812680084688464, 8.907431733525671294691563440852, 10.18862344987889430753876633374, 11.10447479282326342678257779648
