| L(s) = 1 | − 1.38i·2-s − 2.82·3-s + 0.0791·4-s − 0.518i·5-s + 3.91i·6-s + i·7-s − 2.88i·8-s + 4.98·9-s − 0.719·10-s + 1.62i·11-s − 0.223·12-s + 1.38·14-s + 1.46i·15-s − 3.83·16-s − 1.94·17-s − 6.90i·18-s + ⋯ |
| L(s) = 1 | − 0.980i·2-s − 1.63·3-s + 0.0395·4-s − 0.232i·5-s + 1.59i·6-s + 0.377i·7-s − 1.01i·8-s + 1.66·9-s − 0.227·10-s + 0.489i·11-s − 0.0645·12-s + 0.370·14-s + 0.378i·15-s − 0.958·16-s − 0.472·17-s − 1.62i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7829061626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7829061626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.38iT - 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 0.518iT - 5T^{2} \) |
| 11 | \( 1 - 1.62iT - 11T^{2} \) |
| 17 | \( 1 + 1.94T + 17T^{2} \) |
| 19 | \( 1 + 2.49iT - 19T^{2} \) |
| 23 | \( 1 - 9.14T + 23T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 + 5.79iT - 31T^{2} \) |
| 37 | \( 1 + 10.2iT - 37T^{2} \) |
| 41 | \( 1 - 4.20iT - 41T^{2} \) |
| 43 | \( 1 - 0.997T + 43T^{2} \) |
| 47 | \( 1 + 4.51iT - 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 - 6.20iT - 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8.37iT - 67T^{2} \) |
| 71 | \( 1 - 5.19iT - 71T^{2} \) |
| 73 | \( 1 + 11.8iT - 73T^{2} \) |
| 79 | \( 1 - 0.982T + 79T^{2} \) |
| 83 | \( 1 - 8.91iT - 83T^{2} \) |
| 89 | \( 1 + 12.0iT - 89T^{2} \) |
| 97 | \( 1 + 4.42iT - 97T^{2} \) |
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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
−9.570288164892078687426894770327, −9.045652102150496466606251015197, −7.41224846783908450236089566670, −6.80103298269003369019819667319, −5.95481767578546647017531980252, −5.02071258205162245109516099370, −4.30232525608653631266408214877, −2.92161641936650031078282337252, −1.67007901761001347956875416871, −0.45722400072079209930621035243,
1.28590327515186817096668358902, 3.10337058263254258376276487097, 4.66837641349912179036535291760, 5.23583917556723101285428435972, 6.09611956241746658190449975232, 6.74204802630680576833367999802, 7.21325336437978578603127167572, 8.266194719357682427709745627442, 9.252046306266889972758135987299, 10.51864269153722037578705329024
