| L(s) = 1 | − 0.618i·2-s − i·3-s + 1.61·4-s − 0.618·6-s − 2i·7-s − 2.23i·8-s − 9-s − 3·11-s − 1.61i·12-s + i·13-s − 1.23·14-s + 1.85·16-s + 0.236i·17-s + 0.618i·18-s − 6.70·19-s + ⋯ |
| L(s) = 1 | − 0.437i·2-s − 0.577i·3-s + 0.809·4-s − 0.252·6-s − 0.755i·7-s − 0.790i·8-s − 0.333·9-s − 0.904·11-s − 0.467i·12-s + 0.277i·13-s − 0.330·14-s + 0.463·16-s + 0.0572i·17-s + 0.145i·18-s − 1.53·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.312411005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312411005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.618iT - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 0.236iT - 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 7.61iT - 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 4.70T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 9.61iT - 43T^{2} \) |
| 47 | \( 1 + 9.23iT - 47T^{2} \) |
| 53 | \( 1 + 6.76iT - 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 - 9.18iT - 67T^{2} \) |
| 71 | \( 1 + 1.09T + 71T^{2} \) |
| 73 | \( 1 + 2.29iT - 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 2.85iT - 97T^{2} \) |
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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
−8.606037261840449960165868714753, −8.088592876974064850635118439011, −7.00607146701682031854145591107, −6.75517523272380675154696923870, −5.78678932456484747606989159406, −4.61971934042264101553154085310, −3.63878777804347991647917846809, −2.55312829957554503856751277856, −1.83194035763247316576318712521, −0.41977842533846591014964943112,
1.91916599832901717469399012794, 2.75666796808375994024581170538, 3.74930003170665451640786100248, 5.09457328755616601330751222291, 5.55092566155764734104371721170, 6.37132880279268762045785523158, 7.27134843999938928957285534591, 8.093174114011152601551294457686, 8.686123507184514160921969591919, 9.588918047029381318857274067614
